Uniform Positive Improvingness, Tail Norm Condition and Spectral Gap

Liming Wu

First Version: March 2001. Second Version: October 2001

Abstract

In this work we study the two recent notions: resolvent-uniform positive improvingness (=RUPI) and the resolvent-tail norm condition (=RTNC), for a Markov operator P or for a C_{0}-semigroup of Markov operators. Various equivalent characterizations of them, both analytical and probabilistical are presented, which generalize and enrich several very recent results. Moreover they allow us to obtain some explicit estimates on spectral gap and on the exponential convergence rate in the general not necessarily symmetric case, in terms of the RUPI coefficient and of the tail norm parameter, refining the characterization of the exponential convergence by Hino [22](2000) and the characterization of spectral gap by Gong and author [16] (2000). These estimates also extend the work of Aida [1](1998) from the particular case of continuous time symmetric hyperbounded Markov semigroups to the general not-necessarily symmetric case and to the discrete time case. For absolute continuous kernels, a weak compactness result is obtained, and the relation between weak Poincaré inequality and essential spectral gap is studied by means of the Nussbaum formula of essential spectral radius due to Weis [51] and de Pagter and Schepp [7]. Several examples and applications are provided.

1 Introduction

1.1 Two recent characterizations of spectral gap

Let π be a nonnegative operator on L^{p}(μ) := L^{p}(E, B, μ; ℝ) where μ is a probability measure on (E, B) with B countably generated, and 1 ≤ p < +∞. π is said to be positive improving, if π1_{A} > 0, μ − a.s. for any measurable subset A ∈ B with μ(A) > 0. Introduce

Definition 1.1. The nonnegative operator π is called uniformly positive improving w.r.t. μ (UPI/μ in short), if

χ_{π}(ε/μ) := inf {⟨1_{A}, π1_{B}⟩_{μ} | μ(A) ∧ μ(B) ≥ ε} > 0, ∀ε > 0.  (1.1)

Definition 1.2. For a positive operator π on L^{p}(μ), we call

Ψ_{π}(L/p, μ) := sup_{0≤f,∥f∥_{p}≤1} ∥πf1_{[πf>L]}∥_{L^{p}(μ)}, ∀L > 0.  (1.2)

tail norm parameter of π, and

∥π∥_{tail(p,μ)} := lim_{L→∞} Ψ_{π}(L/p, μ)  (1.3)

tail norm of π in L^{p}(μ).

Let (P_{t})_{t∈𝕋} be a semigroup of Markov and bounded operators on L^{p}(E, B, μ), where 𝕋 = ℕ (the discrete time case) or ℝ^{+} (the continuous time case). In the later case (P_{t}) is assumed to be strongly continuous (i.e., C_{0}-semigroup). Let L := P − I in the discrete time case where P = P_{1}; and in the continuous time case, L denotes the generator of (P_{t})_{t∈ℝ^{+}}. Let

R_{λ} := ∑_{n=0}^{∞}(λ + 1)^{−n−1}P^{n} or ∫_{0}^{∞} e^{−λt}P_{t}dt, ∀λ > 0

according to 𝕋 = ℕ or ℝ^{+}, be the resolvent which is well defined on L^{∞}. Assume that

(A1) (μ-essential irreducibility) for any A ∈ B with μ(A) > 0, R_{λ}1_{A} > 0, μ − a.e..

(A2) R_{λ}f ∈ L^{p}(μ) for all 0 ≤ f ∈ L^{p}(μ) and for each λ > 0.

We can now present

Theorem 1.1. (due to Hino [22](2000)) Under (A1) and (A2), assume that there exists ψ > 0, μ − a.e. in L^{p′}(μ) (1/p + 1/p′ = 1) such that L^{∗}ψ = 0 (or equivalently α := ψμ is an invariant measure of (P_{t})), where L^{∗} is the adjoint operator of L, . Then the following two properties are equivalent:

(a) (exponential convergence) There exist some δ, C(δ) > 0 such that

∥P_{t}f − α(f)∥_{L^{p}(μ)} ≤ C(δ)e^{−δt}∥f∥_{L^{p}(μ)}, ∀f ∈ L^{p}(μ), t ≥ 0.  (1.4)

(b) the following two conditions are satisfied:

(E) ∀ε > 0, ∃t ∈ 𝕋 such that χ_{P_{t}}(ε/μ) > 0;

(I) there exists some t_{0} ∈ 𝕋 such that ∥π∥_{tail(p,μ)} < 1.

For application of this beautiful and important result, there are two points to be verified: first the assumption of the existence of the invariant measure μ; next how to verify Conditions (E) and (I). Note also that the exponential convergence is not valid even when the spectral gap of (P_{t}) exists (thinking about a finite irreducible Markov matrix which is not aperiodic).

Theorem 1.2. (due to Gong and Wu [16](2000)) Assume (A1) and (A2). The following properties are equivalent:

(a) (existence of spectral gap) λ_{gap}(L) := inf{|λ|; λ ∈ Σ_{L^{p}(μ;ℂ)}(L)\{0}} > 0, where L^{p}(μ; ℂ) := L^{p}(E, B, μ; ℂ) is the complexification of L^{p}(μ), and Σ_{L^{p}(μ;ℂ)}(L) is the spectrum of L in L^{p}(μ; ℂ) (denoted simply by Σ(L) later).

(b) the following two conditions are satisfied:

(RUPI) (Resolvent-UPI) for some λ > 0, the resolvent R_{λ} is UPI/μ;

(RTNC) (Resolvent Tail Norm Condition) ∃λ > 0, ∥λR_{λ}∥_{tail(p,μ)} < 1.

In that case there exists ψ > 0, μ−a.e. in L^{p′}(μ) (1/p+1/p′ = 1) such that L^{∗}ψ = 0.

For comparison notice that (E) (resp. (I)) is stronger than (RUPI) (resp. (RTNC)) (see the text). Moreover under (RUPI)+(RTNC), we get in [16] the decomposition of Hopf and the exponential convergence involving cyclic classes.

The two results above hold also for general positive operators by means of the ground state representation, see [22] and [16]. Theorems 1.1 and 1.2 and ideas behind are crucial to the solution of a conjecture of Gross [20] about the existence of spectral gap for Schrödinger operators on the loop space, by Gong-Röckner-Wu [14].

Those two results motivate us to in further investigate the RUPI and RTNC and to understand what they mean separately.

1.2 Some historical comments

Theorem 1.1 and Theorem 1.2 and the two involved notions: the UPI and the tail norm condition are the outcome of two lines of development.

The UPI (uniform positive improvingness) is introduced by Kusuoka [26] (1992) in his deep studies of Malliavin calculus and named by Aida [1](1998). The UPI property turns out to be crucial in many issues, especially in the infinite dimensional case, illustrated by the very active and recent researches by Aida [1] (1998) and [2] (2000), Mattiau [29](1998), Hino [22](2000) and [23](2000), Röckner and Wang [42](2000), Wang [49],[50] (2000), Gong and the author [16](2000), Gong, Röckner and the author [14] (2000) etc.

For giving a quick look on the UPI, let us recall

Theorem 1.3. Let (P_{t})_{t≥0} be a C_{0}-semigroup of symmetric Markov operators on L^{2}(E, B, μ). The following properties are equivalent:

(i) χ_{P_{t}}(ε/μ) > 0 for each ε > 0 and each t > 0, i.e., P_{t} is UPI for each t > 0;

(ii) The weak spectral gap property (WSG in short) of its associated Dirichlet form (E, 𝔻(E)), i.e., for any sequence (f_{n})_{n≥1} in 𝔻(E), if ∥f_{n}∥_{L^{2}(μ)} ≤ 1, ∫ f_{n}dμ = 0, (∀n ≥ 1), and lim_{n→∞}E(f_{n}, f_{n}) = 0, then f_{n} → 0 in probablity-μ as n → ∞.

(iii) sup_{∥f∥_{∞}≤1} ∥P_{t}f − ⟨f⟩_{μ}∥_{2} → 0 as t goes to infinity;

(iv) ∀ε > 0, there is some t > 0 such that χ_{P_{t}}(ε/μ) > 0.

(v) For any r > 0 there is c(r) > 0 so that

∥f∥_{L^{2}(μ)}^{2} ≤ c(r)E(f, f) + r∥f∥_{L^{∞}(μ)}^{2}

for all f ∈ 𝔻(E) ∩ L^{∞}(μ) with ⟨f⟩_{μ} := ∫_{E} fdμ = 0.

Here (i) ⟹ (ii) is due to Kusuoka [26] (1992)(who proved essentially (iv) ⟹ (ii)). (ii)⇔(iii) is an observation due to Mattieu [29](1998). Aida [1](1998) established (ii)⇔(iv). The equivalence between (iii) and (iv) in full generality (including the non-symmetric case) is established by Hino [22](2000) and that between (iv) and (i) is shown again by Hino [23] (2000)(closing so the proof of the equivalence between (i), (ii), (iii) and (iv)). Finally the equivalence between (ii) and (v) is proved by M.Röckner and F.Y. Wang in [42](2000).

The name of weak spectral gap property (WSGP in short) for (ii) is given by Aida [1](1998) (for the following simple reason : if ”f_{n} → 0 in probability” is replaced by ”f_{n} → 0 in L^{2}(μ)” in the conclusion of (ii), then property (ii) is equivalent to the existence of spectral gap). The importance of the WSGP for the existence of spectral gap is already noted by Kusuoka, Mattieu and Aida. For instance Aida [1] noticed that under the WSGP, if moreover the following F -Sobolev inequality

∫_{E} f^{2}F (f^{2})dμ ≤ aE(f, f) + b, ∀f ∈ 𝔻(E), ∥f∥_{2} = 1  (1.5)

holds for some increasing function F : ℝ^{+} → ℝ^{+} satisfying lim_{t→∞} F (t) = +∞, then the spectral gap exists. Indeed, for any sequence (f_{n}) given in (ii), f_{n} → 0 in probability by the WSGP. But by the inequality above, (f_{n}) is also uniformly integrable in L^{2}. Thus f_{n} → 0 in L^{2}, the existence of spectral gap follows.

Now the reader see that Hino’s Theorem 1.1 improves the above result.

Many of those beautiful equivalences in Theorem 1.3 fail unfortunately either if the symmetry assumption is removed or if in the discrete time case (see the examples in Section 2). Only the equivalence between (iii) (a strong type’s ergodicity) and (iv) remains valid without the symmetry condition and both in the continuous and discrete time case, as proved by Hino [22]. We emphasize that this equivalence is one key to his proof of Theorem 1.1. (For the capacitary positive improvingness, see [59].)

We now turn to an overview on the notion of the tail norm condition such as (I) and (RTNC). The notion of tail norm can be traced back to de Pagter and Schepp [7](1988). Indeed it is easy to see that ([16])

∥π∥_{tail(p,μ)} = inf_{L>0} sup_{0≤f,∥f∥_{L^{p}(μ)}≤1} ∥(πf − L)^{+}∥_{L^{p}(μ)}

= inf_{0≤u∈L^{p}(μ)} sup_{0≤f,∥f∥_{L^{p}(μ)}≤1} ∥(πf − u)^{+}∥_{L^{p}(μ)} (1.6)

where the last quantity is just the measure of non-semicompactness ρ(π), introduced by de Pagter and Schepp [7](1988) in the general setting of Banach lattices. With the help of (1.6), the Nussbaum formula of essential spectral radius (Nussbaum [35]) due to Weis [51] and de Pagter-Schep [7] read as:

Theorem 1.4. (due to Weis [51](1984), de Pagter-Schep [7](1988)) Let πf(x) := ∫_{E} p(x, y)f(y)μ(dy) be a nonnegative absolute continuous kernel operator, bounded on L^{p}(μ) (1 < p < +∞). Then

r_{ess}(π) := sup{|λ|; λ ∈ Σ_{ess}(π)} = lim_{n→∞}(∥π^{n}∥_{tail(p,μ)})^{1/n}.  (1.7)

where Σ_{ess}(π) is the Wolf essential spectrum of π in L^{p}(μ; ℂ).

This result alone is enough to show the importance of the tail norm (the last name, perhaps more appropriate in the setting L^{p}, is given in [16]. It may be also named as measure of non-uniform integrability by following the spirit of de Pagter-Schep). See [51], [7] and the textbook of Meyer-Nieberg [31] for many other results.

However in the infinite dimensional case, the absolute continuity condition in Theorem 1.4 is often not satisfied and the Nussbaum type formula (1.7) becomes false. Indeed for the standard Ornstein-Uhlenbeck semigroup (P_{t}) on the infinite dimensional abstract Wiener space (W, H, μ) (see Section 6), it is hypercontractive on L^{2}(ℝ, μ) by Nelson-Gross, then the r.h.s. of (1.7) for π = P_{1} equals to 0, but the essential spectral radius of P_{1}, same as its spectral gap, is e^{−1}.

The last example shows that condition (I) or (RTNC) is rooted at the notion of hypercontractivity of Nelson and Gross [19]: π is said to be hypercontractive (resp. hyperbounded) on L^{p}(μ) if ∥π∥_{L^{p}→L^{q}} ≤ 1 (resp. < +∞) for some q > p. The hypercontractivity and the related logarithmic Sobolev inequality of Gross are now developed as a wide domain: see Bakry [3] and Gross [21] for survey and abundant references.

For a hyperbounded operator π, we have ∥π∥_{tail(p,μ)} = 0. If this last condition is verified we say that π is uniformly integrable in L^{p}(μ). That notion was introduced by the author [53](1995) and [55](2000) for the characterization of the large deviation principle of Donsker-Varadhan of Markov processes in the τ-topology. Moreover the existence of ground state, Hopf’s decomposition, finiteness of invariant and ergodic measures (Markov case), continuity of spectral radius w.r.t. the strong convergence of operators, the existence of spectral gap (under the irreducibility), the compactness (under the absolute continuity) etc are established for uniformly integrable operators in [55](2000).

To see quickly what means the uniform integrability, let us consider the weak Poincaré inequality below ([15]): there are some r_{0} ≥ 0 and some non-increasing function β : (r_{0}, +∞) → ℝ^{+} such that for all f ∈ L^{2}(μ)

⟨f^{2}⟩_{μ} ≤ rE(f, f) + β(r)⟨|f|⟩_{μ}^{2}, ∀r > r_{0}.  (1.8)

Theorem 1.5. Let (P_{t})_{t≥0} be a C_{0}-semigroup of positive symmetric operators on L^{2}(μ) satisfying (A2), whose quadratic form associated with −L is denoted by (E, 𝔻(E)). The following properties are equivalent:

(i) for each t > 0, P_{t} is uniformly integrable, i.e., ∥P_{t}∥_{tail(2,μ)} = 0;

(ii) for each (or equivalently for some) λ > 0, R_{λ} is uniformly integrable;

(iii) {|f|^{2}; ∥f∥_{2} + ∥Lf∥_{2} ≤ 1} is uniformly integrable;

(iv) {|f|^{2}; ⟨f, f⟩_{μ} + E(f, f) ≤ 1} is uniformly integrable;

(v) the F -Sobolev inequality (1.5) holds;

(vi) the weak Poincaré inequality (1.8) holds for r_{0} = 0.

If moreover P_{t}, t > 0 can be realized as μ-absolute continuous kernel, then they are in further equivalent to

(vii) P_{t} is compact on L^{2} for each t > 0.

Here the equivalence between (i), (ii), (iii), (iv) is established in [55] (Lemma 4.1, Proposition 4.2 and Remarks (4.iii)); and the equivalent descriptions (v) and (vi) are obtained by Gong and Wang [15]. The equivalence with compactness (vii) is contained in [53], [55] and [48].

Theorem 1.6. (due to Gong and Wang [15], and Wang [49](2000)) In the context of Theorem 1.5 together with the absolute continuous condition, then (a) ⇔ (b) holds:

(a) the weak Poincaré inequality (1.8) holds; (b) the Wolf essential spectrum Σ_{ess}(−L) of −L is contained in [1/r_{0}, +∞).

See Wang [49], [50] and Röckner-Wang [42] for diverse interpretations and applications of the weak Poincaré inequality (1.8). In [42] (Theorem 2.1) they proved essentially (1.8) implies condition (I) of Hino.

We use so large space to present those recent known results above for two reasons:

1) to present a picture about the state of art for the reader; 2) they not only will be used in this work but especially serve as source of ideas and questions.

1.3 Purposes and Organization of the paper.

Our main purposes in this work are:

1) to establish various equivalent characterizations of the RUPI in the non-symmetric case, generalizing and enriching Theorem 1.3;

2) to generalize Theorems 1.5 for the resolvent tail norm condition (RTNC) or (I) in the non-symmetric case; and to find the probabilistic meaning of (RTNC);

3) to provide some explicit estimates on spectral gap in terms of RUPI coefficient and tail-norm parameter of resolvent, in the purpose of better understanding Theorem 1.1 and 1.2;

4) to yield infinitesimal type description of the essential spectral gap for continuous time semigroup corresponding to Theorem 1.4 (which can not often be applied because one knows usually the generator L rather than the semigroup itself), extending Theorem 1.6;

5) to apply our results to several examples in the infinite dimensional case.

One of my personal goals in this work is to convince the reader for the interest and the usefulness of the UPI and the tail norm condition, and do some publicity of them both for probabilists and analytists. That’s why we shall keep our presentation as elementary as possible. Only devoted to specialists are the examples in Section 6 and the proof of the probabilistic characterization of (RTNC) in the last section 9.

This paper is organized as follows. In the next section we present our main results concerning different characterizations of the RUPI both for a Markov operator (discrete time case) and for a C_{0}-semigroup of Markov operators (the continuous time case), and both in the invariant measure or non-invariant measure cases.

In Section 3, we study the conditions (I) and (RTNC). Infinitesimal criteria for (RNTC) are proved in Lemma 3.4 and Theorem 3.5, extending Theorem 1.4 for (RTNC) in the general non-symmetric case. And we prove the equivalence between (I) and (RTNC) under the sector condition. More importantly, we show that for essentially irreducible Markov processes, (RTNC) alone implies the exponential convergence in probability of the empirical mean to the mean w.r.t. the invariant measure, and the inverse is valid for reversible Markov processes.

Section 4 is devoted to find some quantitative versions of Theorem 1.1 and Theorem 1.2. More precisely we get some explicit spectral gap estimates by means of RUPI coefficient and tail norm parameter, extending some previous results of Aida [1] for hyperbounded symmetric Markov semigroups.

Section 5 is devoted to the study of absolute continuous kernels. We establish a weak compactness result, extending the classical Rellich-Kondrachov compactness theorem in the theory of Sobolev spaces. As corollaries, we obtain a similar formula as in Wang’s Theorem 1.6(b) for the essential spectral gap of general non-symmetric semigroups.

Several applications and examples are furnished in Section 6:

1) for the second quantification semigroups over an abstract Wiener space we show the equivalence between the RUPI and the spectral gap;

2) for the second quantification semigroups over a Poisson space we extend the beautiful characterization of the UPI by Röckner and Wang [42]; and finally

3) an application to Girsanov semigroups is investigated.

As mentioned before, the remained part of the paper, Section 7, 8 and 9, are devoted to the proof of the results on the RUPI and on the equivalence between (RTNC) and the exponential convergence in probability.

Before our investigation, let’s make several conventions of notation. Throughout this paper for p, q, ⋯ ∈ [1, +∞], p′, q′, ⋯ denote their conjugated numbers (i.e., 1/p + 1/p′ = 1). (E, B) is a measurable space where σ-algebra B is countably generated. The space of real bounded and B-measurable functions will be denoted by bB. We write as usual notation

⟨f⟩_{ν} := ν(f) := ∫_{E} fdν, ⟨f, g⟩_{ν} := ∫_{E} fgdν.

for a measurable function f and a nonnegative measure ν (if this takes sense of course). And

B_{ν}^{+} := {B ∈ B; ν(B) > 0}, L_{+}^{p}(ν) := {f ∈ L^{p}(ν)| f ≥ 0, ν − a.e., ⟨f⟩_{ν} > 0}.

Throughout this paper, the probability measure μ plays a special role in notation. With some abuse we remove ”μ” in the notations of the objects related with μ. For instance, we use the simplified notations L^{p}, ∥ · ∥_{p} to denote L^{p}(μ) and its norm; χ_{π}(ε) instead of χ_{π}(ε/μ), UPI and RUPI instead of UPI/μ, RUPI/μ; Ψ_{π}(L/p) and ∥ · ∥_{tail(p)} (or simply Ψ_{π}(L), ∥ · ∥_{tail}) instead of Ψ_{π}(L/p, μ), ∥ · ∥_{tail(p,μ)}; etc.

For a linear operator π on L^{p}(μ), it is called

• nonnegative, if πf ≥ 0, μ − a.s. for each f ∈ L_{+}^{p}(μ);

• positive, if it is nonnegative and not trivial (i.e., π ≠ 0);

• sub-Markov, if it is nonnegative and π1 ≤ 1, μ − a.e.;

• Markov, if it is nonnegative and π1 = 1, μ − a.e..

2 Characterizations of the RUPI

2.1 Notations and some rudiments

For any positive and bounded operator π on L^{p}(E, B, μ) =: L^{p}(μ) =: L^{p} (p ∈ [1, +∞)), since B is countably generated, there exists a positive finite kernel π̃(x, dy) on (E, B) such that π̃f(x) := ∫_{E} f(y)π̃(x, dy) coincide with πf(x) for μ − a.e.x for any f ∈ bB, the space of bounded and B-measurable real functions on E (see [10]). Such kernel realization verifies μπ̃(·) := ∫_{E} μ(dx)π̃(x, ·) ≪ μ, and is unique up to μ-a.e. (for x) equivalence.

For a positive finite kernel N(x, dy) : E × B → [0, +∞) verifying μN ≪ μ, the UPI coefficient χ_{N}(ε/μ) given in (1.1) is well defined (as an extended number in [0, +∞]) , too. Similarly N is said to be UP I/μ, if χ_{N}(ε/μ) for any ε > 0.

Of course the UPI property of N becomes meaningful only if χ_{N}(ε/μ) (∀ϵ > 0) is finite, e.g. if N1 ∈ L^{1}(μ). Moreover χπ̃(ε/μ) = χ_{π}(ε/μ) if π̃ is any kernel realization of a positive operator π on L^{p}(E, μ).

Given a positive bounded operator π or a C_{0}-semigroup (π_{t})_{t∈ℝ^{+}} on L^{p}(μ). Set

R_{λ} := ∑_{n=0}^{∞}(1 + λ)^{−n−1}π^{n}, or := ∫_{0}^{∞} e^{−λt}π_{t}dt,

It is a well defined bounded operator for λ strictly greater than the largest real spectral point of π − I or the generator of (π_{t}) (see Lemma 3.3). Similarly

Definition 2.1. We say that a positive bounded operator π on L^{p}(μ) is RUPI/μ (=resolvent uniformly positive improving), if R(λ) is bounded and UPI/μ for some sufficiently large λ.

For a C_{0}-semigroup of bounded and positive operators (π_{t})_{t≥0} on L^{p}(E, μ), we say that it is RUPI /μ, if R(λ) is bounded and UPI/μ for some λ.

Recall that for a semigroup of Markov operators (P_{t})_{t∈𝕋}, its ergodicity w.r.t. the invariant measure μ (i.e., if f ∈ bB satisfies P_{t}f = f for all t ∈ 𝕋, then f is constant μ-a.s) is equivalent to the positive improving property of its resolvent ([40] or [55], Lemma 3.5). The same holds for a C_{0}-semigroup of Markov operators (P_{t})_{t≥0}. That’s why the RUPI is a quite natural extension of the ergodicity.

Before presenting our main results on the RUPI, we give several elementary lemmas. We begin with the following simple but very efficient criterion of perturbation type

Lemma 2.1. ( [22], [16], [16]) Assume that the nonnegative finite kernel N(x, dy) satisfies μN ≪ μ, N1 ∈ L^{1}(μ) and is UPI/μ. If p(x, y) is B × B-measurable and μ⊗N(dx, dy) := μ(dx)N(x, dy)-a.e. positive, then the kernel pN := p(x, y)N(x, dy) is UPI w.r.t. any probability measure ν which is equivalent to μ.

Proof. For the self-containedness we reproduce the proof in [17]. As ν ∼ μ, for any ε > 0, there is a(ε) > 0 so that for each A ∈ B with ν(A) > ε, μ(A) > a(ε).

Let h(x, y) := p(x, y)·(dν/dμ)(x). For any measurable A, B with ν(A)∧ν(B) > ε and for any b > 0,

∫∫_{E×E} 1_{A}(x)1_{B}(y)p(x, y)N(x, dy)ν(dx) = ∫∫_{E×E} 1_{A}(x)1_{B}(y)h(x, y)N(x, dy)μ(dx)
≥ b ∫∫_{E×E} 1_{A}(x)1_{B}(y) (1 − 1_{[h(x,y)≤b]}) N(x, dy)μ(dx)
≥ b [χ_{N}(a(ε)/μ) − μ ⊗ N(h ≤ b)] .

Since h(x, y) > 0, μ ⊗ N − a.e. (and μ ⊗ N is a finite measure), then for b sufficiently small, μ ⊗ N(h ≤ b) < χ_{N}(a(ε)/μ). Hence

χ_{pN}(ε/ν) ≥ b [χ_{N}(a(ε)/μ) − μ ⊗ N(h ≤ b)] > 0.

The same proof leads to

Lemma 2.2. Let ε > 0. Assume that the nonnegative kernel N(x, dy) verifies N1 ∈ L^{1}(μ) and χ_{N}(ε/μ) > 0. If p(x, y) is B × B-measurable and μ ⊗ N(dx, dy) := μ(dx)N(x, dy)-a.e. positive, then

χ_{pN}(ε/μ) > 0.

The difference between Lemmas 2.1 and 2.2 is: in Lemma 2.1 we assume that χ_{N}(ε/μ) > 0 for every ϵ > 0, but only for some given ε > 0 in Lemma 2.2.

This criterion (Lemma 2.1 and Lemma 2.2) is applicable to two important classes of semigroups: Feynman-Kac semigroups and the Girsanov semigroups (given by the Girsanov formula). See [16], [17] for such applications.

Lemma 2.3. Let π, (π_{t}) be respectively a positive bounded operator and a C_{0}- semigroup of positive and bounded operators on L^{p}(μ). Then the following properties are equivalent:

(a) π is RUPI/μ;

(b) for each ε > 0, there exists λ such that R_{λ}1 ∈ L^{1}(μ) and χ_{R_{λ}}(ε/μ) > 0;

(c) for all λ ∈ ℝ so that R_{λ}1 ∈ L^{1}(μ), R_{λ} is UPI/μ.

Proof. 1) Single operator case. Let π̃(x, dy) be a positive kernel realization of π and define similarly G(λ, π̃) := ∑_{k=0}^{∞}(1 + λ)^{−k−1}π̃^{k} (usual multiplication of kernels). Notice that the positive measures R(λ, π̃)(x, ·), λ ∈ ℝ, G(λ, π̃)1 ∈ L^{1}(μ)}, are all equivalent for every x ∈ E, and finite for μ − a.e. x ∈ E. Then the equivalence between (a), (b), (c) follows from Lemma 2.1 and Lemma 2.2.

2) continuous time case. For each t ≥ 0 let π̃_{t} be a kernel realization of π_{t}. For each A ∈ B, (t, x) → π̃_{t}(x, A) is jointly measurable on (t, x) w.r.t. the completion B(ℝ^{+}) × B by dt ⊗ μ. (Warning: In general (π̃_{t})_{t≥0} is not a semigroup of kernels: the realization of a semigroup of kernels of (π_{t}) is a very delicate matter.)

Let R̃_{λ}(x, A) := ∫_{0}^{∞} e^{−λt}π̃_{t}(x, A)dt ∈ [0, +∞]. For μ − a.e. x ∈ E, the positive measures R̃_{λ}(x, ·), λ ∈ ℝ so that R̃_{λ}1 ∈ L^{1}(μ), are all equivalent and finite. Then the desired equivalence follows again from Lemma 2.1 and Lemma 2.2.

We turn now to present a very general situation where the RUPI is satisfied.

Proposition 2.4. Let (π)_{t∈𝕋}, 𝕋 = ℕ or ℝ^{+} be a semigroup of finite positive kernels on (E, B) (i.e., (π_{s}π_{t})(x, B) := ∫_{E} π_{s}(x, dy)π_{t}(y, B) = π_{s+t}(x, B) for all (x, B) ∈ E × B), bounded on L^{p}(E, μ) (necessarily μπ_{t} ≪ μ). Assume that it is strongly continuous if 𝕋 = ℝ^{+}. Consider the resolvent type kernel R_{λ}(x, B) := ∑_{n=0}^{+∞}(1 + λ)^{−n−1}π^{n}(x, B) or := ∫_{0}^{∞} e^{−λt}π_{t}(x, B)dt according to 𝕋 = ℕ or ℝ^{+}, for each λ ∈ ℝ (which may be infinite).

If (π_{t}) is irreducible w.r.t. μ, i.e., for some λ ∈ ℝ, ∀B ∈ B_{μ}^{+},

R_{λ}(x, B) ∈ (0, +∞], for all x ∈ E  (2.1)

then π := π_{1} (if 𝕋 = ℕ) or (π_{t})_{t∈ℝ^{+}} is RUPI/μ.

Proof. We prove it only in the continuous time case.

Fix λ_{0} such that R_{λ_{0}} is bounded in L^{p}(μ). R(λ_{0}) satisfies again (2.1), because the family of measures {R_{λ}(x, ·); λ ∈ ℝ} are equivalent, for each x ∈ E fixed. Now recall a crucial known fact (see [34], p.16): there exist 1 ≤ n_{0} ∈ ℕ, s(x) ∈ bB with μ(s) > 0 and a probability measure ν ≪ μ such that

[R(λ_{0})]^{n_{0}}(x, B) ≥ s(x) · ν(B), ∀(x, B) ∈ E × bB.

Hence

[R(λ_{0})]^{n_{0}+2}(x, B) ≥ (R(λ_{0})s)(x) · (νR(λ_{0}))(B)

By the irredubility assumption, R(λ_{0})s > 0 everywhere over E, and νR(λ_{0}) is equivalent to μ. Hence the kernel at the r.h.s. above is UPI/μ by Lemma 2.1. Consequently [R(λ_{0})]^{n_{0}+2} is UPI/μ. Thus R(λ_{0}) is UPI/μ by Lemma 2.1, for R(λ_{0})(x, ·) is equivalent to [R(λ_{0})]^{n_{0}+2}(x, ·) for each x ∈ E.

When π admits a μ−a.e. positive eigenvector ϕ associated with its spectral radius R(π) (such eigenvector is called ground state, see Lemma 3.2 for quite general criteria for the existence of ground state), the ground state representation

P f → \frac{1}{ϕ}π(ϕf)

is a Markov operator (i.e., P is positive and P 1 = 1) on L^{p}(ν := ϕ^{p}μ). The similar reduction can also be done in the continuous time case. So we are brought to the Markovian case. That’s why our study in this section will mainly focus to the Markovian case in which the probabilistic intuition can guide our investigation.

The framework and objects of this section are as follows. Given a semigroup of Markov operators (P_{t})_{t∈𝕋} on L^{p}(ν), where 𝕋 = ℕ or ℝ^{+}, for clarity we shall separate the studies of its RUPI/ν into four cases:

1) 𝕋 = ℕ (discrete time) and ν is an invariant measure μ;

2) 𝕋 = ℕ, but ν is a non-invariant measure;

3) 𝕋 = ℝ^{+} and ν is an invariant measure μ;

4) 𝕋 = ℝ^{+}, but ν is a non-invariant measure;

2.2 RUPI of a Markov operator: invariant measure case

The following is our first main result in this paper. In its statement two notations μ, ν are used to denote the same invariant measure μ (a little strange I should concede). But this will simplify largely the later statements in the non-invariant measure case ν ≠ μ.

Let L_{0}^{p}(μ) := {f ∈ L^{p}(μ); ⟨f⟩_{μ} := ∫_{E} fdμ = 0}.

Theorem 2.5. Let P be a Markov kernel acting on L^{p}(E, ν) where ν = μ and μP = μ (invariant probability measure), p ∈ (1, +∞). Let P_{μ}^{∗} be the adjoint operator of P w.r.t. μ. The following properties are equivalent.

(P1) P is RUP I/μ.

(P2) The symmetrized Markov operator P^{σ} := \frac{P +P_{μ}^{∗}}{2} acting on L^{2}(μ) is RUP I/μ.

(P3) For any q ∈ (1, +∞], there exists a non-decreasing strictly positive function Θ_{q} : (0, ∞) → (0, +∞) such that for all f ∈ L_{0}^{q}(μ) ⋂ L^{∞},

Θ_{q} (\frac{∥f∥_{L^{1}(μ)}}{∥f∥_{L^{q}(μ)}} ) ∥f∥_{L^{1}(μ)}^{2} ≤ ⟨f, (I − P)f⟩_{μ},  (2.2)

(P4) For each r > 0, there exists c(r) > 0 so that for all f ∈ L_{0}^{2}(μ),

(⟨|f|⟩_{μ})^{2} ≤ r⟨f^{2}⟩_{μ} + c(r)⟨f, (I − P)f⟩_{μ}.  (2.3)

(P4b) For any sequence (f_{n}) ⊂ L_{0}^{2}(μ) with ∥f_{n}∥_{L^{2}(μ)} ≤ 1 such that E^{σ}[f_{n}] → 0, then ∥f_{n}∥_{L^{1}(μ)} → 0. Here E^{σ}(f, g) := ⟨f, (I − P^{σ})g⟩_{μ} is the symmetrized Dirichlet form associated with P^{σ}, and

E^{σ}[f] := E^{σ}(f, f) = ⟨f, (I − P)f⟩_{μ}.

(P5) For each r > 0, there exists c(r) > 0 so that for all f ∈ L_{0}^{∞}(μ),

(⟨|f|⟩_{μ})^{2} ≤ r∥f∥_{∞}^{2} + c(r)E^{σ}[f].  (2.4)

(P5b) For any sequence (f_{n}) ⊂ L_{0}^{∞}(μ) with ∥f_{n}∥_{∞} ≤ 1 such that E^{σ}[f_{n}] → 0, then ∥f_{n}∥_{L^{1}(μ)} → 0.

(P6) For each r > 0, there exists c(r) > 0 so that for all f ∈ L_{0}^{∞}(μ),

(∥f∥_{L^{p}(μ)})^{2} ≤ r∥f∥_{∞}^{2} + c(r)E^{σ}[f].  (2.5)

(P6b) For any sequence (f_{n}) ⊂ L_{0}^{∞}(μ) with ∥f_{n}∥_{∞} ≤ 1 such that E^{σ}[f_{n}] → 0, then ∥f_{n}∥_{L^{p}(μ)} → 0.

(P7) For each r > 0, there exists c(r) > 0 so that for all f ∈ L_{0}^{p}(μ),

∥f∥_{L^{1}(μ)} ≤ r∥f∥_{L^{p}(μ)} + c(r)∥(1 − P)f∥_{L^{1}(μ)}.  (2.6)

(P7b) For any sequence (f_{n}) ⊂ L_{0}^{p}(μ) such that ∥f_{n}∥_{L^{p}(μ)} ≤ 1 and ∥(I−P)f_{n}∥_{L^{1}(μ)} → 0, then ∥f_{n}∥_{L^{1}(μ)} → 0.

(P8) For each r > 0, there exists c(r) > 0 so that for all f ∈ L^{p}(ν) with ⟨f⟩_{μ} = 0,

∥f∥_{L^{1}(ν)} ≤ r∥f∥_{L^{p}(ν)} + c(r)∥(1 − P)f∥_{L^{p}(ν)}.  (2.7)

(P8b) For any sequence (f_{n}) ⊂ L^{p}(ν) such that ⟨f_{n}⟩_{μ} = 0, ∥f_{n}∥_{L^{p}(ν)} ≤ 1 and ∥(I − P)f_{n}∥_{L^{p}(ν)} → 0, then ∥f_{n}∥_{L^{1}(ν)} → 0.

(P9) For each r > 0, there exists c(r) > 0 so that for all f ∈ L^{∞}(ν) with ⟨f⟩_{μ} = 0,

∥f∥_{L^{1}(ν)} ≤ r∥f∥_{∞} + c(r)∥(1 − P)f∥_{L^{∞}(ν)}.  (2.8)

(P9b) For any sequence (f_{n}) ⊂ L^{∞}(ν) such that ⟨f_{n}⟩_{μ} = 0, ∥f_{n}∥_{∞} ≤ 1 and

∥(I − P)f_{n}∥_{L^{∞}(ν)} → 0, then ∥f_{n}∥_{L^{1}(ν)} → 0.

(P10) (uniform ergodic theorem) As n goes to infinity,

∥\frac{1}{n} ∑_{k=0}^{n−1} P^{k} − μ(·)∥_{L^{∞}→L^{1}(ν)} := sup_{f:∥f∥_{∞}≤1} ∥\frac{1}{n} ∑_{k=0}^{n−1} P^{k}f − ⟨f⟩_{μ}∥_{L^{1}(ν)} ⟶ 0. (2.9)

(P10b) (uniform ergodic theorem) For each q ∈ (p, +∞], as n goes to infinity,

∥\frac{1}{n} ∑_{k=0}^{n−1} P^{k} − μ(·)∥_{L^{q}(μ)→L^{p}(μ)} ⟶ 0.  (2.10)

(P11) (uniform ergodic theorem of resolvent) As λ decreases to 0,

∥λR_{λ} − μ(·)∥_{L^{∞}→L^{1}(ν)} ⟶ 0.  (2.11)

(P11b) (uniform ergodic theorem of resolvent) For each q ∈ (p, +∞], as λ → 0+,

∥λR_{λ} − μ(·)∥_{L^{q}(μ)→L^{p}(μ)} ⟶ 0.  (2.12)

Remarks (2.i): Since P is RUPI iff P_{μ}^{∗} is RUPI, we can duplicate again the properties above by using P^{∗} instead of P. With all these equivalence relations, the reader see that the RUPI is a quite natural notion and has many consequences among which the most striking are:

• the uniform ergodic theorems in (P10b), (P11b)

• the weak spectral gap (WSG in short) inequalities (2.2), (2.6) and (2.7) and the corresponding weak spectral gap properties (=WSGP) (Pkb) for k = 4, 7, 8.

Moreover the conditions such as WSGPs in (P6b), (P9b) and WSG inequalities in (P6), (P9) etc, being weaker than the others (only in apparearance), may be used in applications for verifying the RUPI.

Remarks (2.ii): Consider condition (E) in Theorem 1.1 and

(Eb) for some n ≥ 1, P^{n} is UPI.

It is obvious that (Eb) ⟹ (E) ⟹ (RUP I). Hino [22] prove in full generality the equivalence between condition (E) and the following important issue,

∥P^{n} − ⟨·⟩_{μ}∥_{L^{∞}→L^{1}(ν)} → 0.  (2.13)

(this is stronger than the uniform ergodic theorems in Theorem 2.5). In the continuous time symmetric Markov case, this equivalence was at first discovered by Mattiau [29] and Aida [1], as emphasized in the Introduction.

Our remark here is that (2.13) (⇔ condition (E)) is equivalent to the α-mixingness for a Markov operator P w.r.t. its invariant measure μ.

Indeed let ℙ_{μ} be the unique stationary law on (E^{ℤ}, B^{ℤ}) such that its coordinates (X_{n})_{n∈ℤ} is a stationary Markov process with marginal law μ and transition probability P. Recall the coefficients of α-mixingness

\begin{aligned}α_{P} (n) & := sup_{A,B∈B} |⟨1_{A}, P^{n}1_{B}⟩_{μ} − μ(A)μ(B)| \\ α_{ℙ_{μ}}(n) & := sup \{|ℙ_{μ}(A ∩ B) − ℙ_{μ}(A)ℙ_{μ}(B)|; A ∈ σ(X_{k}; k ≤ 0), B ∈ σ(X_{k}; k ≥ n)\} \end{aligned}

We have

\frac{1}{4}∥P^{n} − ⟨·⟩_{μ}∥_{∞,1} ≤ α_{P} (n) ≤ α_{ℙ_{μ}}(n) ≤ ∥P^{n} − ⟨·⟩_{μ}∥_{∞,1}  (2.14)

which gives us the equivalence between (2.13) and the α-mixingness. Although this should be well known, we give its proof for its simplicity. Indeed, only the first inequality needs a proof. For it, noting that

sup_{∥g∥_{∞}≤1} ∫ fgdμ = ∥f∥_{1} = ∫ f · sgn(f)dμ ≤ 2 sup_{A} |∫ 1_{A}fdμ|

where sgn(f) := 1_{f>0} − 1_{f<0}, we have for any f ∈ L^{∞}(μ) with ∥f∥_{∞} ≤ 1

∥P^{n}f − ⟨f⟩_{μ}∥_{1} ≤ 2 sup_{A} |⟨1_{A}, P^{n}f − ⟨f⟩_{μ}⟩_{μ}|

= 2 sup_{A} |⟨(P^{n})^{∗}1_{A} − μ(A), f⟩_{μ}|

≤ 2 sup_{A} ∥(P^{n})^{∗}1_{A} − μ(A)∥_{1}

≤ 4 sup_{A} sup_{B} |⟨(P^{n})^{∗}1_{A} − μ(A), 1_{B}⟩_{μ}| = 4α_{P} (n).

The equivalence between Hino’s condition (E), (2.13) and the α-mixingness does not damage the novelty of the notion UPI. I understand this for two reasons: 1) the UPI coefficient χ_{π}(ε/ν) is defined for any positive operator π (not necessarily only for Markov operators), and even for Markov operators, it is well defined and useful w.r.t. some not-necessarily invariant measure μ; 2) the UPI coefficient is more ”crude” and then easier to calculate than the α-mixingness coefficient α_{P} (n).

It is a surprise (for us) that the necessary and sufficient conditions in Theorem 2.5 fail for characterizing Hino’s condition (E) or the stronger (Eb): for instance for a μ-irreducible Markov operator P, the RUPI holds always (by Proposition 2.4) and then, all properties such as the WSGPs and the weak spectral gap inqualities in Theorem 2.5 are valid, but the condition (E) or (2.13) is not valid once if P is not aperiodic (by Revuz [40] or Nummelin [34]).

Remarks (2.iii): All functions c(r) appearing in the inequalities in Theorem 2.5 can be explicitly given in terms of the UPI coefficient χ_{R_{λ}} of the resolvent R_{λ} in the actual invariant measure case, see Section 7. Here we present the convergence rate in the uniform ergodic theorem:

Proposition 2.6. Assume that P is a Markov kernel with μP = μ. Letting

χ^{(R)}(ε) := sup_{λ>0} χ_{λ^{2}R_{λ}}(ε/μ)  (2.15)

then for any q ∈ (1, +∞] and for all f ∈ L_{0}^{q}(μ)

χ^{(R)} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1} ≤ ∥(I − P)f∥_{1}.  (2.16)

In particular if P is RUPI, then for all n ≥ 1,

∥\frac{1}{n} ∑_{k=0}^{n−1} P^{k} − μ(·)∥_{L^{∞}→L^{1}(μ)} ≤ h^{−1} (\frac{2}{n})  (2.17)

where h(ε) := ε · χ^{(R)}(ε/8) and h^{−1}(r) := inf{ε ∈ (0, 1); h(ε) > r} (convention: inf ∅ := +∞).

χ^{(R)}(ε) defined in (2.15) will be called RUPI coefficient. For comparison note that as R_{λ} ≥ (λ + 1)^{−2}P,

χ^{(R)}(ε) ≥ sup_{λ>0} λ^{2}(λ + 1)^{−2} · χ_{P} (ε) = χ_{P} (ε).  (2.18)

The rate estimate (2.17) is new and explicit, and may produce the correct convergence rate in the uniform ergodic theorem (2.9): 1/n (reached once if P has a spectral gap), as shown by the

Example 1. Assume that the state space E is finite, P = (P (i, j))_{i,j∈E} is a Markov matrix with the invariant measure μ charging all elements. Condition (E) is equivalent to (Eb), which is the famous condition for P^{n}(x, ·) → μ(·) in the Perron-Frobenius theorem. But the RUPI means simply the irreducibility in the actual finite case.

Let μ_{min} := min_{i∈E} μ(i). Then χ^{(R)}(ε) ≥ χ^{(R)}(μ_{min}) for all ε > 0. Hence by (2.17),

∥\frac{1}{n} ∑_{k=0}^{n−1} P^{k} − μ(·)∥_{L^{∞}→L^{1}(μ)} ≤ \frac{2}{nχ^{(R)}(μ_{min})}, ∀n ≥ 1.

It gives the correct order 1/n. But this estimate becomes unpratical when the cardinal #E of E is too large (see Saloff-Coste [43] for references).

More often however, the rate estimate (2.17) is far from being of correct order: indeed if μ is diffuse (i.e., has no atom), χ^{(R)}(ε) = ε^{2} in the most mixing case that P · = μ(·), then h^{−1}(2/n) is of order n^{−1/3}, which is far from being the correct order 1/n. This non-sharpness shows that either our argument given in Section 7 is not sufficiently refined, or for producing sharp rate in this ergodic theorem one should combine (or replace) the RUPI coefficient χ^{(R)} with (by) other parameters (we believe that the last reason is more important). Nevertheless the rate estimate (2.17) in terms of the RUPI coefficient χ^{(R)} is universal: the RUPI is also a necessary condition for this uniform ergodic theorem by Theorem 2.5, and it is very useful in the infinite dimensional case.

2.3 RUPI of a Markov operator: non-invariant measure case

For a given Markov kernel P its invariant measure μ is not always known, but often known to exist. So let us consider a reference probability measure ν ∼ μ such that P : L^{p}(ν) → L^{p}(ν) is bounded. which ones of the properties in Theorem 2.5 remain equivalent with μ substituted by ν ?

The reader can guess the answer with the strange notations in Theorem 2.5:

Theorem 2.7. Let 1 < p < +∞. Assume that P is a Markov kernel, bounded on L^{p}(E, ν) (ν is a probability measure) and μ = ψν is an invariant probability measure with 0 < ψ ∈ L^{p′}(ν) (the invariance of μ ⇔ P_{ν}^{∗}ψ = ψ).

Then the RUPI of P w.r.t. ν is equivalent to each one of the properties in Theorem 2.5 with now ν different from μ.

Remarks (2.iv): Several striking properties in Theorem 2.5 stated with μ, if replaced by ν, are perhaps no longer necessary to the RUPI: for instance the properties where the symmetrized Dirichlet form E^{σ} is involved. But the most important loss is: we have no longer explicit extimate of c(r) in the WSG inequalities, nor estimate of convergence rate in the uniform ergodic theorem (unlike in Proposition 2.6).

2.4 RUPI of a Markov semigroup

We pass now to the continuous time case. Unlike in the discrete time case, we shall state in an unified way our result both for the invariant measure μ case and the non-invariant measure ν case:

Theorem 2.8. Let (P_{t})_{t≥0} be a C_{0}-semigroup of Markov operators on L^{p}(ν) where 1 < p < +∞. Assume that there is some a.e. positive ψ ∈ L^{p′}(ν) such that L^{∗}ψ = 0. (P_{t}) is also C_{0} on L^{r}(μ) for each r ∈ [1, +∞). Let 𝔻_{p,ν}(L) (resp. 𝔻_{p,μ}(L)) be the domain of the generator L of (P_{t}) in L^{p}(ν) (resp. L^{p}(μ)).

The following properties are equivalent:

(C1) (P_{t}) is RUPI/ν.

(C2) The dual semigroup ((P_{t})_{μ}^{∗}) (w.r.t. μ) is RUPI/μ.

(C3) For any q ∈ (1, +∞], there exists some strictly positive non-decreasing function Θ_{q} : (0, 1) → (0, +∞) such that for all non-zero f ∈ L_{0}^{q}(μ) ⋂ 𝔻_{1,μ}(L),

Θ_{q} (\frac{∥f∥_{L^{1}(μ)}}{∥f∥_{L^{q}(μ)}} ) ∥f∥_{L^{1}(μ)} ≤ ∥Lf∥_{L^{1}(μ)}.  (2.19)

(C4) For each r > 0, there exists c(r) > 0 so that for all f ∈ 𝔻_{p,ν}(L) with ⟨f⟩_{μ} = 0,

∥f∥_{L^{1}(ν)} ≤ r∥f∥_{L^{p}(ν)} + c(r)∥Lf∥_{L^{p}(ν)}.  (2.20)

(C4b) For any sequence (f_{n}) ⊂ 𝔻_{p,ν}(L) such that ⟨f_{n}⟩_{μ} = 0, ∥f_{n}∥_{L^{p}(ν)} ≤ 1 and ∥Lf_{n}∥_{L^{p}(ν)} → 0, then ∥f_{n}∥_{L^{1}(ν)} → 0.

(C5) For each r > 0, there exists c(r) > 0 so that for all f ∈ 𝔻_{p,ν}(L) with ⟨f⟩_{μ} = 0,

∥f∥_{L^{p}(ν)} ≤ r∥f∥_{L^{∞}(ν)} + c(r)∥Lf∥_{L^{p}(ν)}.  (2.21)

(C5b) For any sequence (f_{n}) ⊂ 𝔻_{p,ν}(L) such that ⟨f_{n}⟩_{μ} = 0, ∥f_{n}∥_{L^{∞}(ν)} ≤ 1 and ∥Lf_{n}∥_{L^{p}(ν)} → 0, then ∥f_{n}∥_{L^{p}(ν)} → 0.

(C6) For each r > 0, there exists c(r) > 0 so that for all f ∈ L^{∞}(ν) ⋂ 𝔻_{p,ν}(L) with

⟨f⟩_{μ} = 0,

∥f∥_{L^{1}(ν)} ≤ r∥f∥_{∞} + c(r)∥Lf∥_{∞}.  (2.22)

(C6b) For any sequence (f_{n}) ⊂ L^{∞} such that ⟨f_{n}⟩_{μ} = 0, ∥f_{n}∥_{∞} ≤ 1 and ∥Lf_{n}∥_{L^{∞}(ν)} → 0, then ∥f_{n}∥_{L^{1}(ν)} → 0.

(C7) (uniform ergodic theorem) As T goes to infinity,

∥\frac{1}{T} ∫_{0}^{T} P_{t}dt − μ(·)∥_{L^{∞}(ν)→L^{1}(ν)} ⟶ 0.  (2.23)

(C7b) (uniform ergodic theorem) For each q ∈ (p, +∞], as T goes to infinity,

∥\frac{1}{T} ∫_{0}^{T} P_{t}dt − μ(·)∥_{L^{q}(μ)→L^{p}(μ)} → 0.  (2.24)

(C8) (uniform ergodic theorem of resolvent) As λ decreases to 0,

∥λR(λ, L) − μ(·)∥_{L^{∞}→L^{1}(ν)} ⟶ 0.  (2.25)

(C8b) (uniform ergodic theorem of resolvent) For each q ∈ (p, +∞], as λ → 0+,

∥λR(λ, L) − μ(·)∥_{L^{q}(μ)→L^{p}(μ)} ⟶ 0.  (2.26)

Remarks (2.v). When ν is the invariant measure μ, all coefficients c(r) appearing in the inequalities above can be calculated explicitly in terms of the RUPI coefficient

χ^{(R)}(ε) := sup_{λ>0} χ_{λ^{2}R_{λ}}(ε/μ).

Moreover we have the same estimation of the convergence rate in the ergodic theorem (C7) in the invariant measure case, as in Proposition 2.6:

Proposition 2.9. Assume that (P_{t}) is a Markov C_{0}-semigroup with μP_{t} = μ, which is RUPI. Let χ^{(R)}(ε) as defined above. Then for every q ∈ (1, +∞],

χ^{(R)} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1} ≤ ∥Lf∥_{1}, ∀f ∈ L_{0}^{q}(μ) ⋂ 𝔻_{1,μ}(L)  (2.27)

and for all t > 0,

∥\frac{1}{t} ∫_{0}^{t} P_{s}ds − μ(·)∥_{L^{∞}→L^{1}(μ)} ≤ h^{−1} (\frac{2}{t})  (2.28)

where h(ε) := εχ^{(R)}(ε/8) and h^{−1}(r) := inf{ε > 0; h(ε) > r} (inf ∅ := +∞).

Remarks (2.vi). The careful reader see immediately an essential difference between the discrete time and continuous time case: the Dirichlet form type’s WSG inequalities and the associated WSGPs in Theorems 2.5, 2.7 do not appear in Theorem 2.8. They are in reality false under the RUPI in the continuous time case, as shown by

Example 2. ([58]) Consider a linear damping Hamilton system Z_{t} := (x_{t}, y_{t}) (position-velocity) satisfying the Ito SDE in ℝ^{2},

dx_{t} = y_{t}dt, dy_{t} = σdB_{t} − (cy_{t} + bx_{t})dt

where c > 0 is the damping coefficient, b > 0, (B_{t}) is the standard Brownian Motion and σ^{2} > 0 measures the random perturbation strength. This is a Markov Gaussian process with the unique invariant measure given by the Gaussian measure

μ(dx, dy) = exp (−\frac{2c}{σ^{2}} (\frac{1}{2}|y|^{2} + \frac{b}{2}|x|^{2})) dxdy

and the transition kernel P (Z_{t} ∈ dz′|Z_{0} = z) (for each t > 0) is a Gaussian absolutely continuous measure with C^{∞}-density p_{t}(z, z′) > 0 everywhere over ℝ^{2} × ℝ^{2}. Hence P_{t} is UPI for each t > 0 by Lemma 2.1.

But the counterpart of form-type inequality (v) in Theorem 1.3 (due to Röckner-Wang) given below

∥f∥_{L^{1}(ℝ^{2},μ)}^{2} ≤ r∥f∥_{L^{∞}(ℝ^{2},μ)}^{2} + c(r)⟨−Lf, f⟩_{μ},

is false. Indeed, one easily show that for f ∈ C^{2}(ℝ^{2}) with compact support,

⟨−Lf, f⟩_{μ} = \frac{σ^{2}}{2} ∫∫_{ℝ^{2}} (\frac{∂}{∂y} f(x, y))^{2} μ(dx, dy).

Therefore the above inequality can not hold for all f(x, y) = f(x) depending only on x. We leave to the reader the verification that the counterparts of (Pk), (Pkb) for k=3, 4, 5, 6 in Theorem 2.5 are all false for this example.

This example tells us that in order to have form-type inequalities under RUPI in the continuous time case, further assumption is required.

Our answer is given by the following result which extends Theorem 1.3 (except the UPI in Theorem 1.3.(i)),

Theorem 2.10. Let (P_{t}) be a C_{0}-semigroup of Markov operators on L^{2}(μ) with μP_{t} = μ. Assume the strong sector condition for the generator (L, 𝔻_{2}(L)):

∃κ ≥ 1 : ⟨−Lf, g⟩_{μ} ≤ κ\sqrt{⟨−Lf, f⟩_{μ} · ⟨−Lg, g⟩_{μ}}, ∀f, g ∈ 𝔻_{2}(L)  (2.29)

Then the RUPI of (P_{t})_{t≥0} is equivalent to, besides those in Theorem 2.8, each one of (E1) for any q ∈ (1, +∞], there exists some strictly positive non-decreasing function

Θ_{q} : (0, 1) → (0, +∞) such that for all f ∈ L_{0}^{q}(μ) ⋂ 𝔻_{2}(L),

Θ_{q} (\frac{∥f∥_{L^{1}(μ)}}{∥f∥_{L^{q}(μ)}} ) ∥f∥_{L^{1}(μ)}^{2} ≤ ⟨−Lf, f⟩_{μ}.  (2.30)

(E2) For each q ∈ (1, +∞] and for any r > 0, there exists c(r) > 0 so that for all f ∈ L_{0}^{q}(μ) ⋂ 𝔻_{2}(L),

∥f∥_{L^{1}(μ)}^{2} ≤ r∥f∥_{L^{q}(μ)}^{2} + c(r)⟨−Lf, f⟩_{μ}.  (2.31)

(E3) For each p ∈ [1, +∞) and for any r > 0, there exists c(r) > 0 so that for all f ∈ L_{0}^{∞}(μ) ⋂ 𝔻_{2}(L)

∥f∥_{L^{p}(μ)}^{2} ≤ r∥f∥_{L^{∞}(μ)}^{2} + c(r)⟨−Lf, f⟩_{μ}.  (2.32)

(E4) ∥P_{t} − μ(·)∥_{L^{∞}→L^{1}(μ)} ⟶ 0 as t goes to infinity.

(E5) For any +∞ ≥ q > p ≥ 1, ∥P_{t} − μ(·)∥_{L^{q}(μ)→L^{p}(μ)} ⟶ 0 as t goes to infinity.

(E6) (α-mixingness) α(t) = sup_{A,B∈B} |⟨1_{A}, P_{t}1_{B}⟩ − μ(A)μ(B)| ⟶ 0 as t → ∞.

(E7) (condition (E) of Hino) For each ε > 0, ∃t > 0: χ_{P_{t}}(ε) > 0.

See its proof in Section 7 for the explicit expression of c(r) appearing in the precedent inequalities. Below we give an estimate on the convergence rate in (E4).

Proposition 2.11. Assume the sector condition (2.29) and the RUPI of (P_{t}) w.r.t. the invariant measure μ (the same context as in Theorem 2.10). Then there exists some constant M > 0 depending only on κ, such that

∥P_{t} − μ(·)∥_{L^{∞}→L^{1}(μ)} ≤ h^{−1} (\frac{M}{t} )  (2.33)

where h(ε) := εχ^{(R)}(ε/8) and h^{−1}(r) := inf{ε > 0; h(ε) > r} (inf ∅ := +∞).

3 Resolvent tail norm condition: characterization and probabilistic meaning

In this section we study the resolvent tail norm condition: its equivalent descriptions in terms of the (analytical) weak Poincaré inequalities and the (probabilistical) exponential convergence in probability.

In this section (P_{t}) is no longer necessarily a Markov semigroup unlike in Section 2, but a general semigroup of positive and bounded operators on L^{p}(E, B, μ) =: L^{p}(μ) =: L^{p} (this is not only for generality but for the following reason: the ground state representation does not affect UPI or RUPI by Lemma 2.1, but influences very much condition (I) or (RTNC) especially when the ground states are not positive a.e. (see however [22])). When (P_{t}) is Markov, its invariant measure is denoted sometimes by μ, sometimes by α, but explicitly stated for avoiding the confusion.

3.1 RTNC: some basic properties

See (1.2) and (1.3) in the Introduction for the definition of the tail norm parameter Ψ_{π}(L/p, μ) and the tail norm ∥π∥_{tail(p,μ)}. We shall remove p, μ in the previous notations (related with μ, see our convention fixed in Introduction).

Note that π is bounded iff ∥π∥_{tail} < +∞; π is uniformly integrable in L^{p}(μ) iff ∥π∥_{tail} = 0.

The following facts are established in Gong and the author [16]:

Lemma 3.1. ([16]) Define ∥π∥_{tail(L^{∞})} := lim sup_{μ(A)→0} ∥1_{A}π∥_{∞}.

For positive and bounded operators π and π_{1}, π_{2} on L^{p}(μ) with 1 ≤ p < +∞, the followings hold (with ∥ · ∥_{tail} = ∥ · ∥_{tail(p,μ)}):

(a) Let B_{p}^{+}(1) := {f ∈ L^{p}(μ); f ≥ 0, μ − a.e., ∥f∥_{p} ≤ 1}.

∥π∥_{tail} = lim sup_{μ(A)→0} sup_{g∈B_{p}^{+}(1)} ∥1_{A}πg∥_{p} = lim sup_{μ(A)→0} sup_{g∈B_{p}^{+}(1)} ∥π(g1_{A})∥_{p}

= lim sup_{μ(A),μ(B)→0} sup_{g∈B_{p}^{+}(1)} ∥1_{B}π(g1_{A})∥_{p}; (3.1)

(b) ∥π∥_{tail(L^{p}(μ))} = ∥π^{∗}∥_{tail(L^{p′}(μ))} where π^{∗} is the adjoint operator;

(c) ∥π_{1} + π_{2}∥_{tail} ≤ ∥π_{1}∥_{tail} + ∥π_{2}∥_{tail} and ∥λπ∥_{tail} = λ∥π∥_{tail}, ∀λ ≥ 0;

(d) ∥π_{1}π_{2}∥_{tail} ≤ ∥π_{1}∥_{tail} · ∥π_{2}∥_{tail};

(e) If π_{1}f ≤ π_{2}f for all f ∈ L_{+}^{p} (denoted by π_{1} ≤ π_{2}), then ∥π_{1}∥_{tail} ≤ ∥π_{2}∥_{tail}.

Remarks (3.i) The parts (b),(c),(d),(e) are easy consequences of (3.1). This lemma tells us that ∥π∥_{tail} possesses many properties of the operator norm ∥π∥_{p} := ∥π∥_{L^{p}→L^{p}}. But there is a very important property of ∥ · ∥_{p} which is not valid for ∥ · ∥_{tail}:

if π_{n}f → πf in L^{p} for any f ∈ L^{p}, then ∥π∥_{p} ≤ lim inf_{n} ∥π_{n}∥_{p} (Banach’s uniform boundedness theorem); but in general ∥π∥_{tail} ≤ lim sup_{n} ∥π_{n}∥_{tail} is not valid, as seen for the

Example 3. Let E = ℕ and μ a probability measure on ℕ charging all points of ℕ, and π_{n}f(x) = 1_{x≤n}f(x). Obviously ∥π_{n}∥_{tail} = 0, and π_{n} → I strongly. But ∥I∥_{p} = 1.

From now on we assume always 1 < p < +∞ in this Section (except the contrary expression).

Definition 3.1. Given a bounded positive operator P or a C_{0}-semigroup (P_{t})_{t∈ℝ^{+}} of bounded positive operators on L^{p}(μ) such that 0 is a spectral point with the largest real part in the spectrum Σ(L) of L (which is P − I or the generator of (P_{t})) in L^{p}(μ). We say that satisfies the resolvent tail norm condition (RTNC in short), if for some λ > 0,

∥λR_{λ}∥_{tail} < 1

where R_{λ} := (λ − L)^{−1} for λ > 0 is the resolvent.

The following lemma shows two useful consequences of the RTNC:

Lemma 3.2. ([16])

(a) For a Markov and bounded operator P on L^{p}(μ), if ∥λR_{λ}∥_{tail} < 1 for some λ > 0, then the spectral radius r_{sp}(P) = 1;

(b) (existence of ground state) In the context of Definition 3.1, assume the (RTNC). Then there exists 0 ≤ ϕ ∈ L^{p}(μ), 0 ≤ ψ ∈ L^{p′}(μ) with ∥ϕ∥_{p} = ∥ψ∥_{L^{p′}(μ)} = 1 such that P_{t}ϕ = ϕ and P_{t}^{∗}ψ = ψ for all t ∈ 𝕋.

Remark (3.ii): Hino’s condition (I) in Theorem 1.1 is stronger than the RTNC λR_{λ} with λ > 0 sufficiently small. This is an easy consequence of Lemma 3.1.(d) (or by Hino [22], Propositions 2.5 and 2.6). Indeed let us check it in the continuous time case. We have,

lim sup_{λ→0+} ∥R_{λ}∥_{tail} = lim sup_{λ→0+} ∥ ∫_{0}^{∞} e^{−λt}P_{t}∥_{tail}

= lim sup_{λ→0+} ∥ ∑_{k=0}^{∞} ∫_{0}^{t_{0}} e^{−λ(kt_{0}+s)}P_{kt_{0}+s}∥_{tail}

≤ lim sup_{λ→0+} sup_{0≤s≤t_{0}} ∥P_{s}∥_{p} ∑_{k=0}^{∞} (e^{−λt_{0}}∥P_{t_{0}}∥_{tail})^{k}

= sup_{0≤s≤t_{0}} ∥P_{s}∥_{p} · \frac{1}{1 − ∥P_{t_{0}}∥_{tail}} .

But Hino’s condition (I) is strictly stronger than the RTNC given in Definition 3.1 at least in the continuous time case. This is shown by the Example 4.(uniform motion on the circle). Let (θ_{t}x := e^{it}x) be the uniform motion on the unit circle S of the complex plan, equipped with the normalized Lebesgue measure μ = dx. Let P_{t}f(x) := f(θ_{t}x). Its resolvents R_{λ} for λ > 0 are compact on L^{2}(S, dx), then ∥R_{λ}∥_{tail} = 0 for any λ > 0 (then satisfying the RTNC). But it is easy to see that ∥P_{t}∥_{tail} = 1 for all t ≥ 0, i.e., the condition (I) of Hino is violated.

To clarify the definition 3.1 of RUPI, we must understand what means “0 is a spectral point with largest real part of L”. To this end we present the following lemma which should be known.

Lemma 3.3. (a) Given a positive operator P, r_{sp}(P) ∈ Σ(P). In particular 0 is a spectral point with largest real part of L := P − I iff r_{sp}(P) = 1.

(b) Given a C_{0}-semigroup of positive operators (P_{t})_{t≥0}, we have

sup{Reλ; λ ∈ Σ(L)} = inf{λ ∈ ℝ| ∫_{0}^{∞} e^{−λt}P_{t}fdt ∈ L^{p}, ∀f ∈ L_{+}^{p}(μ)} ∈ Σ(L) (3.2)

and for λ > sup{Reλ; λ ∈ Σ(L)} ∈ Σ(L),

(λ − L)^{−1}f = ∫_{0}^{∞} e^{−λt}P_{t}fdt convergent in L^{p}, ∀f ∈ L^{p}  (3.3)

In particular the following properties are equivalent:

(i) 0 is a spectral point with largest real part of the generator L;

(ii) 0 ∈ Σ(L) and each λ > 0 belongs to the resolvent set of L;

(iii) If λ > 0, then ∫_{0}^{∞} e^{−λt}P_{t}fdt ∈ L^{p}, ∀f ∈ L_{+}^{p}(μ); but if λ < 0, ∫_{0}^{∞} e^{−λt}P_{t}fdt ∉ L^{p} for some f ∈ L_{+}^{p}(μ).

Warning: In the continuous time case, it may happens that

ω := lim_{t→∞}(1/t) log ∥P_{t}∥_{p} > sup{Reλ; λ ∈ Σ(L)}.

(There is known counter-example.) This shows the subtleness of (3.2).

Proof. The first claim in Part (a) is well known, see [31]. The second immediately follows from the first.

(b). Denote the two quantities in (3.2) respectively by λ_{1} and λ_{2}. We show at first λ_{1} ∈ Σ(L).

For any a > ω := lim_{t→∞} \frac{1}{t} log ∥P_{t}∥_{p}, a belongs to the resolvent set of L and (a−L)^{−1} = ∫_{0}^{∞} e^{−at}P_{t}dt. By the spectral mapping theorem, Σ(R_{a}) = {(a−λ)^{−1}; λ ∈ Σ(L)}. In further by the first claim in (a), r_{sp}(R_{a}) = sup{|z|; z ∈ Σ(R_{a})} ∈ Σ(R_{a}). Consequently there is some λ_{0} ∈ Σ(L) ⋂ ℝ such that

\frac{1}{a − λ_{0}} = sup \{\frac{1}{|a − λ|}; λ ∈ Σ(L)\} .

Such λ_{0} is obviously independent of a > ω. Let us show λ_{0} = λ_{1}. Indeed if in contrary λ_{1} > λ_{0} (λ_{1} ≥ λ_{0} being trivial), then there would be some λ ∈ Σ(L) with Reλ > λ_{0}. That is in contradition with the equality above for a > ω sufficiently large. Therefore λ_{1} = λ_{0} ∈ Σ(L).

Now we turn to establish λ_{1} = λ_{2}. Notice that for each f ∈ L_{+}^{p}(μ), g ∈ L_{+}^{p′}(μ),

h_{1}^{(f,g)}(z) = ⟨(z − L)^{−1}f, g⟩_{μ}, h_{2}^{(f,g)}(z) = ∫_{0}^{∞} e^{−zt}⟨P_{t}f, g⟩_{μ}dt

are respectively analytic on D_{k} := {z ∈ ℂ; Rez > λ_{k}}, k = 1, 2 (the analyticity of h_{2}^{(f,g)}(z) follows from the fact that ∫_{0}^{N} e^{−zt}⟨P_{t}f, g⟩_{μ}dt converges to ∫_{0}^{∞} e^{−zt}⟨P_{t}f, g⟩_{μ}dt uniformly (on z) over the compacts of D_{2}). They coincide for real z > ω.

Moreover for any b < λ_{1}, there exist always (f, g) ∈ L_{+}^{p} × L_{+}^{p′} such that h_{1}^{(f,g)}(z) has no analytical extension on the larger semi-plan {z; Rez > b} (than D_{1}), for (z − L)^{−1} has a singularity at z = λ_{1} ∈ Σ(L).

Similarly but a little more subtle, for any b < λ_{2}, there exist always (f, g) ∈ L_{+}^{p} × L_{+}^{p′} such that h_{1}^{(f,g)}(z) has no analytical extension on the larger semi-plan {z; Rez > b} (than D_{2}). The reason is the following: if the Laplace transformation h_{2}^{(f,g)}(z) of the nonnegative measure ⟨P_{t}f, g⟩_{μ}dt, well defined on D_{2}, could be extended analytically on {z; Rez > b} for some b < λ_{2}, then ∫_{0}^{∞} e^{−λt}⟨P_{t}f, g⟩_{μ}dt < +∞ for all (f, g) ∈ L_{+}^{p}×L_{+}^{p′} and for all λ > b. That would imply ∫_{0}^{∞} e^{−λt}P_{t}fdt ∈ L^{p}, ∀f ∈ L_{+}^{p}, for all λ > b (by Banach’s uniform boundedness theorem). This is in contradiction with the definition of λ_{2}.

For the reasons above, we see that the maximal demi-plans on which h_{1}^{(f,g)}, h_{2}^{(f,g)} allow analytical extension for all (f, g) ∈ L_{+}^{p} × L_{+}^{p′}, which are respectively D_{1}, D_{2}, are the same. Thus λ_{1} = λ_{2} and h_{1}^{(f,g)}(λ) = h_{2}^{(f,g)}(λ) for all λ > λ_{1} = λ_{2} and for all (f, g) ∈ L_{+}^{p} × L_{+}^{p′}. The proof of (3.2) and (3.3) is finished.

Finally the equivalence between (i), (ii) and (iii) are easy consequence of (3.2).

3.2 RTNC: characterizations

A more important advantage of the RTNC over the condition (I) is the infinitesimal criteria (T4),(T5) in Theorem 3.5 below (particularly in the continuous time case, in which usually only the generator L is known, not (P_{t})). To this end, we begin with the key

Lemma 3.4. Assume that

(A2) each λ > 0 belongs to the resolvent set of L in L^{p} (recalling that L = P − I or the generator of (P_{t})).

(It is equivalent to (A2) written in Introduction by Lemma 3.3.) Let r_{0} be the infimum of those r > 0 such that there exists β(r) ≥ 0, such that the following weak Poincaré inequality holds for all f ∈ 𝔻_{p}(L) (the domain of L in L^{p}(μ)),

∥f∥_{p} ≤ r∥Lf∥_{p} + β(r)∥f∥_{L^{1}(μ)}  (3.4)

(convention: r_{0} := +∞ if no such r exists). Then

lim_{λ→0+} ∥R_{λ}∥_{tail(p,μ)} = r_{0}.  (3.5)

Proof. For (3.5), we establish at first

lim inf_{λ→0+} ∥R_{λ}∥_{tail} ≥ r_{0}.

Without loss of generality we assume that the l.h.s. above is finite. Pick out a sequence (λ_{n}) such that the previous lim inf equals to lim_{n→∞} ∥R_{λ_{n}}∥_{tail}. As this last limit is assumed to be finite, lim_{n→∞} ∥λ_{n}R_{λ_{n}}∥_{tail} = 0, then we can assume in further that ∥λ_{n}R_{λ_{n}}∥_{tail} < 1 for all n.

Given L > 0 and for any positive operator π and g ∈ L^{p}(μ), we have by denoting c := ∥g∥_{p}, g̃ = g/c,

∥πg∥_{p} ≤ ∥1_{[|πg|>Lc]}πg∥_{p} + ∥1_{[|πg|≤Lc]}πg∥_{p}
≤ c · ∥1_{[π|g̃|>L]}π|g̃|∥_{p} + (Lc)^{1/p′} · ∥πg∥_{L^{1}(μ)}^{1/p}
≤ Ψ_{π}(L) · ∥g∥_{p} + (L∥g∥_{p})^{1/p′} · ∥πg∥_{L^{1}(μ)}^{1/p}.

For any λ > 0 and f ∈ 𝔻_{p}(L), applying the above inequality to π = λR_{λ} (which is bounded and positive in L^{p}(μ)) and g = \frac{1}{λ}(λ − L)f, we get by noting λR_{λ}g = f,

\begin{aligned}∥f∥_{p} & ≤ Ψ_{λR_{λ}}(L) · \frac{1}{λ}∥(λ − L)f∥_{p} + (\frac{L}{λ}∥(λ − L)f∥_{p})^{1/p′} · ∥f∥_{L^{1}(μ)}^{1/p} \\ & ≤ (Ψ_{λR_{λ}}(L) + \frac{ε}{p′} ) · \frac{1}{λ}∥(λ − L)f∥_{p} + \frac{1}{p} (\frac{L}{ε})^{p/p′} · ∥f∥_{L^{1}(μ)}, \end{aligned}  (3.6)

where ε > 0 is arbitrary and the inequality

xy ≤ \frac{1}{ε^{p/p′}} x^{p} + \frac{ε}{p′} y^{p′}, ∀x, y ≥ 0

is used. Therefore for λ = λ_{n}, choosing L, ε > 0 so that Ψ_{λR_{λ}}(L) + \frac{ε}{p′} < 1 (possible by our assumption done above), from (3.6) we derive (3.4) for any f ∈ 𝔻_{p}(L), with

\begin{aligned}r & = r(λ, L, ε; p) = \frac{1}{λ} (1 − Ψ_{λR_{λ}}(L) − \frac{ε}{p′} )^{−1} (Ψ_{λR_{λ}}(L) + \frac{ε}{p′} ); \\ β(r) & = β(λ, L, ε; p) = (1 − Ψ_{λR_{λ}}(L) − \frac{ε}{p′} )^{−1} \frac{1}{p} (\frac{L}{ε})^{p/p′} . \end{aligned}  (3.7)

Letting at first ε → 0+, next L → +∞ and finally λ = λ_{n} → 0+, we see that the limit of r(λ, L, ε; p) is equal to

lim_{n→∞} \frac{∥R_{λ_{n}}∥_{tail}}{1 − ∥λ_{n}R_{λ_{n}}∥_{tail}} = lim_{n→∞} ∥R_{λ_{n}}∥_{tail}.

Thus r_{0} ≤ lim_{n→∞} ∥R_{λ_{n}}∥_{tail} = lim inf_{λ→0+} ∥R_{λ}∥_{tail}.

We now turn to prove the inverse inequality

lim sup_{λ→0+} ∥R_{λ}∥_{tail} ≤ r_{0}

in (3.5). Without loss of generality we assume that r_{0} < +∞.

For any r > r_{0} fixed, the inequality (3.4) holds for some β = β(r) > 0. Applying it to f = R_{λ}g where g ∈ L^{p}(μ) (λ > 0) and noting −LR_{λ}g = g − λR_{λ}g, we get

∥R_{λ}g∥_{p} ≤ r∥(I − λR_{λ})g∥_{p} + β∥R_{λ}g∥_{L^{1}(μ)}

where it follows that, as soon as λ < 1/r,

∥R_{λ}g∥_{p} ≤ \frac{r}{1 − rλ}∥g∥_{p} + \frac{β}{1 − rλ}∥R_{λ}g∥_{L^{1}(μ)}.

Using the fact (a) recalled in Lemma 3.1, we obtain

∥R_{λ}∥_{tail} ≤ \frac{r}{1 − rλ} + \frac{β}{1 − rλ} lim sup_{μ(B)→0} sup_{g∈B_{p}^{+}(1)} ∥R_{λ}(g1_{B})∥_{L^{1}(μ)}

where B_{p}^{+}(1) := {g ∈ L^{p}(μ) ; g ≥ 0, ∥g∥_{L^{p}(μ)} ≤ 1}. Notice that as μ(B) → 0,

sup_{g∈B_{p}^{+}(1)} ∥R_{λ}(g1_{B})∥_{L^{1}(μ)} = sup_{g∈B_{p}^{+}(1)} sup_{∥h∥_{∞}≤1}⟨R_{λ}(g1_{B}), h⟩_{μ}

≤ sup_{g∈B_{p}^{+}(1)}⟨R_{λ}^{∗}1, g1_{B}⟩_{μ}

≤ ∥1_{B}R_{λ}^{∗}1∥_{p′} ⟶ 0

where the third “≤” follows by the Hölder’s inequality, and the last convergence follows by the fact that R_{λ}^{∗}1 ∈ L^{p′}(μ). Therefore

∥R_{λ}∥_{tail} ≤ \frac{r}{1 − rλ}.  (3.8)

Letting λ → 0+ and noting that r > r_{0} is arbitrary, we obtain the desired inverse inequality. So (3.5) is established.

Theorem 3.5. Assume (A2). We have (T 1) ⇔ (T 2) ⇔ (T 3) ⇔ (T 2) ⇔ (T 4) ⇒ (T 5):

(T1) For some λ > 0, ∥λR_{λ}∥_{tail} < 1.

(T2) lim_{λ→0+} ∥λR_{λ}∥_{tail} = 0.

(T3) lim_{λ→0+} ∥R_{λ}∥_{tail} = r_{0} < +∞.

(T4) (weak Poincaré inequality) There are r_{0} ∈ ℝ^{+} and some non-increasing function β : (r_{0}, +∞) → ℝ^{+} such that the weak Poincaré inequality (3.4) holds for all r > r_{0}. (the r_{0} here coincide with that in (T3)).

(T5) For any sequence (f_{n})_{n∈ℕ} ⊂ B_{p}(1) ⋂ 𝔻_{p}(L) (B_{p}(1) being the unit ball of L^{p}(μ)) such that ∥Lf_{n}∥_{p} → 0, (|f_{n}|^{p})_{n∈ℕ} is uniformly integrable.

If moreover lim inf_{λ→0+} ∥λR_{λ}∥_{L^{p}(μ)→L^{p}(μ)} < +∞, then (T 5) ⇒ (T k) for k = 1, 2, 3, 4.

Proof. (T1) ⟹ (T4): This is established in the proof of (3.4) with the coefficients r, β(r) given by (3.7).

(T4) ⟹ (T3). It follows by Lemma 3.4.

(T3) ⟹ (T2) ⟹ (T1): trivial.

Then (Tk), k=1,2,3,4 are equivalent.

(T3) ⟹ (T5): For any sequence (f_{n})_{n∈ℕ} ⊂ B_{p}(1) ⋂ 𝔻_{p}(L) such that ∥Lf_{n}∥_{p} → 0, for its uniform integrability in L^{p}(μ), we have only to prove that

lim sup_{μ(A)→0} sup_{n≥0} ∥1_{A}f_{n}∥_{p} = 0.

Fix for moment N ≥ 1 and λ > 0. Set

h_{n}^{λ} = −R_{λ}Lf_{n} = f_{n} − λR_{λ}f_{n}.

We have

lim sup_{μ(A)→0} sup_{n≥N} ∥1_{A}h_{n}^{λ}∥_{p} ≤ sup_{n≥N} ∥Lf_{n}∥_{p} · lim sup_{μ(A)→0} sup_{f∈B_{p}(1)} ∥1_{A}R_{λ}f∥_{p}
≤ ε(N) · ∥R_{λ}∥_{tail}

where ε(N) := sup_{n≥N} ∥Lf_{n}∥_{p} goes to zero as N goes to infinity. Consequently using f_{n} = h_{n}^{λ} + λR_{λ}f_{n} and the precedent estimation, we get

lim sup_{μ(A)→0} sup_{n≥0} ∥1_{A}f_{n}∥_{p} = lim sup_{μ(A)→0} sup_{n≥N} ∥1_{A}f_{n}∥_{p}

≤ lim sup_{μ(A)→0} (sup_{n≥N} ∥1_{A}h_{n}^{λ}∥_{p} + sup_{n≥N} ∥1_{A}λR_{λ}f_{n}∥_{p})

≤ ε(N) · ∥R_{λ}∥_{tail} + ∥λR_{λ}∥_{tail}

Letting λ → 0+ and N → ∞, we obtain the desired result.

(T5) ⟹ (T1) (under the extra assumption): Assume in contrary that (T1) were not valid. There would exist a sequence (λ_{n} > 0) decreasing to zero and a double sequence (g_{n,m})_{n,m∈ℕ} ⊂ B_{p}^{+}(1) ⋂ 𝔻_{p}(L), such that (λ_{n}R_{λ_{n}}) are uniformly bounded on L^{p}(μ) (by the extra assumption), and

lim sup_{μ(A)→0} sup_{m≥0} ∥1_{A}λ_{n}R_{λ_{n}}g_{n,m}∥_{p} ≥ \frac{1}{2}, ∀n.  (3.9)

Consequently {f_{n,m} := λ_{n}R_{λ_{n}}g_{n,m}; n, m ∈ ℕ} is bounded in L^{p}(μ). In further

sup_{m≥0} ∥ − Lf_{n,m}∥_{p} = sup_{m≥0} ∥λ_{n}(g_{n,m} − λ_{n}R_{λ_{n}}g_{n,m})∥ → 0

as n goes to infinity. By the assumed (T5), {f_{n,m} = λ_{n}R_{λ_{n}}g_{n,m}; n, m ∈ ℕ} is uniformly integrable in L^{p}(μ), which is in contradiction with (3.9).

Remarks (3.iii). The extra condition lim_{λ→0+} ∥λR_{λ}∥_{p} < +∞ is satisfied in the following particularly important cases:

(i) (P_{t}) is sub-Markov and μP_{t} ≤ μ (i.e., μ is excessive);

(ii) p = 2 and (P_{t}) is symmetric in L^{2}(μ), satisfies (A2).

(In both two cases, ∥λR_{λ}∥_{L^{p}(μ)→L^{p}(μ)} ≤ 1 for all λ > 0.)

Remind the fact that the infinitesimal criteria (T4), (T5) in Theorem 3.5 for RTNC are not sufficient for the condition (I) of Hino, as seen for Example 4.

By (3.7) and (3.8) in the proof of Lemma 3.4 above, we get immediately (except (iv): its equivalence with (i) is contained in Lemma 4.1 of [55].)

Proposition 3.6. Assume (A2). Then the following properties are equivalent:

(i) for all λ > 0, R_{λ} is uniformly integrable, i.e., ∥R_{λ}∥_{tail} = 0;

(ii) for some λ > 0, R_{λ} is uniformly integrable;

(iii) the weak Poincaré inequality (3.4) holds with r_{0} = 0;

(iv) {|f|^{p}; f ∈ 𝔻_{p}(L), ∥f∥_{p} + ∥Lf∥_{p} ≤ 1} is uniformly integrable.

Remarks (3.iv). Part (iii) of Proposition 3.6 extends Theorem 1.5(v) due to Gong and Wang [15]. See [15], [49], [55] for applications of those infinitesimal criteria.

Proposition 3.6 is of no meaning in the discrete time case (unlike Theorem 3.5), because R_{λ} ≥ (λ + 1)^{−1}I is not uniformly integrable except that B is finite up to μ-equivalent classes.

Example 5. Let P be a Markov operator on (E, B) with invariant probability measure μ, such that B is infinite up to μ-equivalent classes.

Then P_{t} := exp(t(P − I)), t > 0 are not uniformly integrable in L^{p}(μ), by Proposition 3.6(iv). And (P_{t}) satisfies the RTNC iff P does (by Theorem 3.5).

This continuous time semigroup is widely used in stochastic algorithms.

Proposition 3.7. Let (P_{t})_{t∈ℝ^{+}} be a C_{0}-semigroup of positive operators on L^{p}(μ) satisfying (A2). If there are r_{0} ∈ ℝ^{+} and a nonincreasing function β(r) on (r_{0}, +∞) such that

⟨f^{p}⟩_{μ} ≤ r⟨−Lf, f^{p−1}⟩_{μ} + β(r)⟨f⟩_{μ}^{p}, ∀0 ≤ f ∈ 𝔻_{p}(L), r > r_{0},  (3.10)

then for all t ≥ 0,

∥P_{t}∥_{tail} ≤ exp (−\frac{t}{r_{0}} ) .  (3.11)

Inversely if p = 2 and (P_{t}) is moreover symmetric, then (3.11) implies (3.10) with p = 2 for all f ∈ 𝔻_{2}(L).

Proof. For any 0 ≤ f ∈ 𝔻_{p}(L) fixed, P_{t}f ∈ 𝔻_{p}(L) and t → P_{t}f is continuously differentiable in L^{p}(μ) and \frac{d}{dt}P_{t}f = LP_{t}f. Hence by (3.10),

\begin{aligned}\frac{d}{dt}⟨(P_{t}f)^{p}⟩_{μ} & = p⟨LP_{t}f, (P_{t}f)^{p−1}⟩_{μ} \\ & ≤ \frac{p}{r} [β(r)⟨P_{t}f⟩_{μ}^{p} − ⟨(P_{t}f)^{p}⟩_{μ}] \end{aligned}

By Gronwall’s inequality, we obtain for all 0 ≤ f ∈ 𝔻_{p}(L),

⟨(P_{t}f)^{p}⟩_{μ} ≤ exp (−\frac{pt}{r}) (⟨f^{p}⟩_{μ} + ∫_{0}^{t} exp (\frac{ps}{r}) \frac{β(r)p}{r}⟨P_{s}f⟩_{μ}^{p}ds) .  (3.12)

This inequality can be extended for all 0 ≤ f ∈ L^{p}(μ): indeed f_{n} := nR_{n}f ≥ 0, and belongs to 𝔻_{p}(L), and f_{n} → f in L^{p}(μ) as n goes to infinity by the strong continuity of (P_{t}).

By (3.12) and Lemma 3.1, we get (as in the derivation of (3.8))

∥P_{t}∥_{tail}^{p} = lim sup_{μ(A)→0} sup_{f∈B_{p}^{+}(1)}⟨[P_{t}(f1_{A})]^{p}⟩_{μ} ≤ exp (−\frac{pt}{r}) .

Therefore (3.11) follows by noting that r > r_{0} is arbitrary.

To show the inverse in the symmetric case, note at first that −L is self-adjoint and nonnegative. Recall the following formula:

the Cauchy semigroup (Q_{t} = exp(−t\sqrt{−L}) associated with (P_{t}), is given by Q_{0} = I and

Q_{t}f := ∫_{0}^{∞} f_{t}(s)P_{s}fds, t > 0

where (f_{t})_{t>0} is a family of probability densities on ℝ^{+} satisfying f_{u} ⋆ f_{v} = f_{u+v} (convolution) for all u, v > 0, and

∫_{0}^{∞} e^{−λs}f_{t}(s)ds = e^{−t\sqrt{λ}}, ∀λ > 0.

(see [60], Chap. IX, (2), (7), p260). Since ∥P_{t}∥_{tail} ≤ e^{−t/r_{0}}, we obtain ∀t > 0,

∥Q_{t}∥_{tail} ≤ ∫_{0}^{∞} f_{t}(s)∥P_{s}∥_{tail}ds ≤ ∫_{0}^{∞} f_{t}(s)e^{−s/r_{0}}ds = e^{−t/\sqrt{r_{0}}}.

Therefore

lim_{λ→0+} ∥ (λ − \sqrt{−L})^{−1} ∥_{tail} ≤ lim_{λ→0+} ∫_{0}^{∞} e^{−λt}∥Q_{t}∥_{tail}dt = \sqrt{r_{0}}.

By Theorem 3.5 and Cauchy-Schwarz, there exists some non-increasing function β(r) on (r_{0}, +∞) such that for all r > r_{0},

⟨f, f⟩ ≤ r⟨\sqrt{−L}f, \sqrt{−L}f⟩ + β(r)[μ(|f|)]^{2}, ∀f ∈ 𝔻_{2}(\sqrt{−L})

which is the desired result, for ⟨\sqrt{−L}f, \sqrt{−L}f⟩ = ⟨−Lf, f⟩ for all f ∈ 𝔻_{2}(L) ⊂ 𝔻_{2}(\sqrt{−L}).

The weak Poincaré inequality of Dirichlet form type (1.8) or (3.11) is not necessary for the RTNC in the non-symmetric case, as shown by the proposition above and Example 4.

When does the RTNC imply the weak Poincaré inequality of form type (3.10) except the symmetric case? This is partially answered in the following

Theorem 3.8. Let p = 2. Let (P_{t})_{t∈ℝ^{+}} be a C_{0}-semigroup of positive operators on L^{2}(μ) satisfying (A2). Assume that the generator L satisfies the sector condition,i.e., there are constant κ ≥ 1 such that for all f, g ∈ 𝔻_{2}(L), ⟨−Lf, f⟩_{μ} ≥ 0 and

⟨−Lf, g⟩_{μ}^{2} ≤ κ⟨−Lf, f⟩_{μ} · ⟨−Lg, g⟩_{μ}  (3.13)

Then the equivalent properties (T_{k}), k=1,2,3,4,5 in Theorem 3.5 are in further equivalent to each of

(T6) (Condition (I)) for some t > 0, ∥P_{t}∥_{tail} < 1;

(T7) (weak Poincaré inequality of form type) there exists some r_{0} ∈ ℝ^{+} and a non-increasing nonnegative function β(r) on (r_{0}, +∞) such that ,

⟨f, f⟩_{μ} ≤ r⟨−Lf, f⟩_{μ} + β(r)[μ(|f|)]^{2}, ∀r > r_{0}, ∀f ∈ 𝔻_{2}(L).

In that case, ∥P_{t}∥_{tail} ≤ exp(−t/r_{0}) for all t ≥ 0, where r_{0} is given by (T7). If moreover (P_{t}) is sub-Markov, then they are equivalent to

(T8) If f_{n} ∈ B_{2}^{+}(1) ⋂ 𝔻_{2}(L) and ⟨−Lf_{n}, f_{n}⟩_{μ} → 0, then {f_{n}^{2}; n ∈ ℕ} is uniformly integrable.

Its proof will be postponed in the last Section 9.

3.3 Probabilistic meaning of the RTNC

In this paragraph let E be a Polish space and B its Borel σ-field. Assume

(A3) (P_{t})_{t∈𝕋} is a semigroup of bounded Markov operators on L^{p}(μ) which satisfies is strongly continuous if 𝕋 = ℝ^{+}. It corresponds to an E-valued Markov processes (Ω, F, (F_{t})_{t∈𝕋}, (X_{t})_{t∈𝕋}, ℙ_{μ}) with initial measure μ, which is càdlàg in the continuous time case.

We write 𝔼^{x}F := 𝔼^{ℙ_{μ}}(F |X_{0} = x) which is well defined for μ − a.e. x ∈ E. For another initial measure γ ≪ μ, we set ℙ_{γ}(·) := ∫_{E} γ(dx)ℙ_{μ}(·|X_{0} = x), under which the initial law of the Markov process is γ, and 𝔼^{γ}(·) := 𝔼^{ℙ_{γ}}(·) = ∫_{E} γ(dx)𝔼^{x}(·).

Consider the occupation measures

\begin{aligned}L_{t} & := \frac{1}{n} ∑_{k=0}^{n−1} δ_{X_{k}}, if 0 < t = n ∈ 𝕋 = ℕ; \\ & := \frac{1}{t} ∫_{0}^{t} δ_{X_{s}}ds, if 0 < t ∈ 𝕋 = ℝ^{+} \end{aligned}

which are random elements in the space of probability measures on (E, B), denoted by M_{1}(E). On M_{1}(E), we shall equipp it with the so called τ-topology, which, by definition, is generated by the linear forms

β ↦ β(f) := ⟨f⟩_{β} := ∫_{E} f(x)dβ(x), f ∈ bB.

Assume the essential irreducibility (A1). If there exists some invariant probability measure α ≪ μ (such measure is unique by Lemma 3.4 in [55], applied with π there equal to λR_{λ}^{∗}), then (P_{t})_{t∈𝕋} is ergodic w.r.t. α (by Lemma 3.5 in [55]). Hence by the ergodic theorem,

L_{t}(f) → α(f), in probability ℙ_{μ}.

This is equivalent to say that for any τ-neighborhood N(α) of α,

lim_{t→∞} ℙ_{μ} (L_{t} ∉ N(α)) = 0.

If (P_{t}) has a spectral gap in L^{p}(μ), one can prove by following the proof of Corollary ??? in [58] that the rate of this convergence in probability is exponential, i.e., for any f : E → ℝ^{d} bounded and measurable, for any δ > 0,

lim_{t→∞} \frac{1}{t} log ℙ_{μ} (|L_{t}(f) − α(f)| > δ) < 0.  (3.14)

This last property turns out to be almost equivalent to the RTNC, as claimed by

Theorem 3.9. Assume (A1), (A2) and (A3).

(a) Assume that for some t_{0} ∈ 𝕋, λ_{0} > 0,

∥P_{t_{0}}λ_{0}R_{λ_{0}}∥_{tail} < 1.  (3.15)

Then there exists some ψ ∈ L^{p′}(μ) such that L^{∗}ψ = 0, ψ > 0, μ − a.e. and ⟨ψ⟩_{μ} = 1. Moreover for any τ-neighborhood N(α) of α := ψ · μ in M_{1}(E),

lim_{t→∞} \frac{1}{t} log sup_{γ∈A(1+ε,L)} ℙ_{γ} (L_{t} ∉ N(α)) < 0, ∀ε > 0, L > 1.  (3.16)

where A(q, L) := {γ ∈ M_{1}(E); γ ≪ μ, ∥dγ/dμ∥_{q} ≤ L}.

(b) Inversely assume that 𝕋 = ℝ^{+}, μ is an invariant measure and (P_{t}) is symmetric w.r.t. μ. If for some initial measure γ verifying

∀A ∈ B_{μ}^{+}, ∃0 < T ∈ 𝕋 : 𝔼^{ν}L_{T} (A) > 0,  (3.17)

such that the exponential convergence in probability (3.14) holds under ℙ_{γ} (instead of ℙ_{μ}) for any f : E → ℝ^{d} bounded and measurable, d ≥ 1, then (P_{t}) satisfies the resolvent tail norm condition (RTNC).

That is the main result of this section. Its proof, depending heavily on the theory of large deviations and on Theorem 3.5(T5), will be left in the last section 9. Here we do some remarks.

Remarks (3.v). This theorem is largely inspired by Theorem 5.1 in [55], where it was proved that

• If ∥P_{t_{0}}R_{λ}∥_{tail} = 0 for some λ, t_{0} ≥ 0, then ℙ_{γ}(L_{t} ∈ ·) satisfies a large deviation principle on (M_{1}(E), τ) with the rate function given by the modified Donsker-Varadhan’s entropy J_{μ}; (see [55] for concise formulation: the extra condition “sup_{0≤t≤1} ∥P_{t}∥_{1} < +∞” in (5.3) there is not required for the level-2 large deviation principle above, but is needed only for the level-3 large deviation principle.)

• If for some 0 < s ∈ 𝕋, P_{s} is μ-essentially irreducible, then the following uniform lower bound of large deviations

lim inf_{t→∞} \frac{1}{t} log inf_{γ∈A} ℙ_{γ} (L_{t} ∈ G) ≥ −inf_{G}J_{μ}, ∀G(τ-open) ⊂ M_{1}(E)

where A is any family of absolute continuous probability measures γ such that {dγ/dμ; γ ∈ A} is (μ-)uniformly integrable.

• In the continuous time and symmetric case, the inverse of (a) is true.

The large deviation principle given in Theorem 5.1 of [55] is stronger than the exponential convergence in probability here, see Corollary 5.6 therein.

Remarks (3.vi). If the RTNC is satified, then the condition in (a) is satisfied for t_{0} = 0. The assumption in (a) allows for the passage to the process-level. More precisely for each 0 < T ∈ 𝕋, consider the Markov process (X_{t}^{(T)} := X_{[t,t+T]}, t ≥ 0) valued in E^{(T)}, which is the product space E^{T} in the discrete time case or 𝔻([0, T], E), the space of E-valued càdlàg functions on [0, T] equipped with the Skorokhod topology. It is càdlàg in E^{(T)} again (left to the reader). Its associated semigroup (resp. resolvent; empirical measures) is denoted by (P_{t}^{(T)}) (resp. R_{λ}^{(T)}; L_{t}^{(T)}).

Assume C := sup_{0≤t≤1} ∥P_{t}∥_{1} < +∞ (as in [55] for the process-level large deviations). We leave to the reader to verify

1) (P_{t}^{(T)}) is a C_{0}-semigroup of Markov operators on L^{p}(E^{(T)}, μ^{(T)}), where μ^{(T)} is the law of (X_{t})_{t∈[0,T]}.

2) If (P_{t}) satisfies (3.15), then for t ≥ T,

∥P_{t_{0}+t}^{(T)}λR_{λ}^{(T)}∥_{tail(p,μ^{(T)})} ≤ C∥P_{t_{0}}λR_{λ}∥_{tail(p,μ)} · ∥P_{t}∥_{L^{p}(μ)→L^{p}(μ)}.

Hence the assumption (3.15) in (a) of Theorem 3.9 is satisfied for (P_{t}^{(T)}) with t_{0} + T instead of t_{0}. Thus we obtain:

Corollary 3.10. Assume (3.15) and sup_{0≤t≤1} ∥P_{t}∥_{1} < +∞. Then for each T ∈ 𝕋, for any multivariant function F : E^{(T)} → ℝ^{d} bounded and measurable, ∀ε > 0,

lim_{t→∞} \frac{1}{t} log sup_{μ∈A(1+ε,L)} ℙ_{γ} (|L_{t}^{(T)}(F) − 𝔼^{α}F (X_{[0,T]})| > δ) < 0, ∀ε, δ > 0, L > 1. (3.18)

Recall that the statistic estimators like L_{t}^{(T)}(F) are widely used in estimations of transition probabilities of Markov processes.

Remarks (3.vii). We dont know whether the necessity of (RTNC) for the exponential convergence in probability is valid in a more general context than the symmetric case.

4 Estimates of spectral gap

We have so far studied separately the RUPI and RTNC. We now combine those two notions to discuss the problem of spectral gap. The main purpose of this section consists in providing explicite estimates of the spectral in terms of the RUPI coefficient and RTNC parameter, refining Theorem 1.1 and 1.2 at the beginning of the paper.

Our results extend in a quite natural way some previous work of Aida [1] for continuous time symmetric Markov semigroups (Aida [2] recently generalizes in further his results to hyperbounded Feynman-Kac semigroups).

The study of spectral gap has a long and glorious history going back to Poincaré. It is out of my ability to even outline its development during almost a century. The interested reader is referred to Chavel [5], M.F. Chen [6], Bakry and Qian [4] etc for spectral gap of the Laplacian or diffusions on Riemannian manifolds; Diagonis and Stroock [11], Saloff-Coste [43] etc for spectral gap on graphes, etc, and for extended references therein.

4.1 Two new proofs of the sufficiency in Theorem 1.2

We at first do some reduction of the problem.

By Lemma 3.2, if ∥λR_{λ}∥_{tail} < 1, then 1 is a spectral point of λR_{λ} with the largest modulus, and there exists ψ > 0 a.e. in L^{p′}(μ) such that λR_{λ}^{∗}ψ = ψ and μ(ψ) = 1. Then L^{∗}ψ = 0 and consequently P_{t}^{∗}ψ = ψ. Observe that α := ψμ is an invariant probability measure of P or (P_{t}). By (A1), P or (P_{t}) is ergodic w.r.t. α and Ker(L^{∗}) = {cψ; c ∈ ℝ}. By the spectral mapping theorem, 0 is a point in Σ(L) with the largest real part.

For the spectral of L near 0, it is enough to show that

∃λ_{0} > 0 : λ_{0}∥f∥_{p} ≤ ∥Lf∥_{p}, ∀f ∈ 𝔻_{p}(L) with α(f) = 0.  (4.1)

In fact, the above inequality implies that the Poisson operator L^{−1}, restricted to the subspace of f ∈ L^{p}(μ) with α(f) = 0, exists and is bounded ([25]). This means exactly that λ = 0 is a pole of order 1 of the resolvent R_{λ} of L( = P − I or the generator of (P_{t})) with the associated eigenprojection given by f → α(f). This is much more than the existence of specral gap.

We give now two proofs of (4.1).

Proof 1. For (4.1) we have only to show that for any sequence (f_{n}) ⊂ L^{p}(μ) with α(f) = 0, if ∥Lf_{n}∥_{p} → 0, then ∥f_{n}∥_{p} → 0.

Indeed by by the RUPI and Theorem 2.7(P8b), f_{n} → 0 in probability. But by the RTNC and Theorem 3.5(T5), {|f_{n}|^{p}; n} is uniformly integrable. Thus f_{n} → 0 in L^{p}(μ) as desired. □

Proof 2. By the RTNC and Theorem 3.5, for any a > r_{0} ∈ ℝ^{+}, there exists b ∈ ℝ^{+} such that

∥f∥_{L^{p}(μ)} ≤ a∥Lf∥_{L^{p}(μ)} + b∥f∥_{L^{1}(μ)}.

By Theorem 2.7(P8) or Theorem 2.8(C4), we get for each r > 0 and for all f ∈ L^{p}(μ) with ⟨f⟩_{α} = 0,

\begin{aligned}∥f∥_{L^{p}(μ)} & ≤ a∥Lf∥_{L^{p}(μ)} + b∥f∥_{L^{1}(μ)} \\ & ≤ [a + bc(r)] · ∥Lf∥_{L^{p}(μ)} + br∥f∥_{L^{p}(μ)} \end{aligned}

Then once if br < 1 (possible as r > 0 can be arbitrary),

∥f∥_{L^{p}(μ)} ≤ \frac{a + bc(r)}{1 − br}∥Lf∥_{L^{p}(μ)}  (4.2)
□

The first proof is more direct. But the second inequality approach illustrates better our strategy for estimating the spectral gap: the usual Poincaré inequality (4.2) is splitted into two inequalities: the weak spectral gap inequality (in Section 2) corresponding to the RUPI, and the weak Poincaré inequality corresponding to the RTNC. In reality this approach can produce some explicit spectral gap estimates. This will be done in the next paragraphs in the invariant measure case.

4.2 Estimates of spectral gap

Speaking of spectral gap of non-symmetric operators, we are obliged to place in the complexification space L^{p}(E, B, μ; ℂ) =: L^{p}(μ; ℂ) =: L_{ℂ}^{p}. Write L_{0}^{p}(μ; ℂ) := {f ∈ L_{ℂ}^{p}; μ(f) = 0}.

Theorem 4.1. Let (P_{t})_{t∈𝕋} be a (C_{0}-) Markov semigroup with invariant measure μ and P = P_{1} if 𝕋 = ℕ. Consider the spectral gap of L (= P − I or the generator) in L^{p}(μ) near 0,

λ_{gap}(p) := inf{|λ|; λ ∈ Σ_{L^{p}(μ;ℂ)}(L)\{0}}

where Σ_{L^{p}(μ;ℂ)}(L) is the spectrum of L in L^{p}(μ; ℂ). Then

λ_{gap}(p) ≥ c(p)λ_{0}(p), λ_{0}(p) := inf ; 0≠f∈L_{0}^{p}(μ) ⋂ 𝔻_{p}(L) \frac{∥Lf∥_{p}}{∥f∥_{p}}  (4.3)

where c(p) is a universal constant with c(2) = 1. λ_{0}(p) = t satisfies for all λ, L > 0,

\frac{Ψ_{λR_{λ}}(L)}{λ}t + (\frac{L}{λ}(λ + t))^{1/p′} (κ_{p}(t))^{1/p} ≥ 1 − Ψ_{λR_{λ}}(L).  (4.4)

where Ψ_{λR_{λ}}(L) = Ψ_{λR_{λ}}(L/p, μ) is the tail norm parameter, defined in (1.2), and

κ_{p}(t) := 1 ∧ inf \{r ∈ (0, 1); χ^{(R)} ((\frac{r}{4})^{p′}) · r > t\}  (4.5)

with χ^{(R)}(r) := sup_{λ>0} χ_{λ^{2}R_{λ}}(r) being the RUPI coefficient.

In particular for p = 2, for any λ, L > 0, r ∈ (0, 1] such that

χ^{(R)} (\frac{r^{2}}{16}) > 0 and Ψ_{λR_{λ}}(L) + \sqrt{Lr} < 1.  (4.6)

it holds that

λ_{gap}(2) ≥ λ_{0}(2)

≥ [rχ^{(R)} (\frac{r^{2}}{16})] ∧ \{\frac{λ}{2 (Ψ(L))^{2}} [2Ψ(L)[1 − Ψ(L)] + Lr − \sqrt{Lr[4Ψ(L) + Lr]}]\}

≥ [rχ^{(R)} (\frac{r^{2}}{16})] ∧ \{\frac{λ}{4 (Ψ(L))^{2}} (−\sqrt{Lr} + \sqrt{Lr + 4Ψ(L) [1 − Ψ(L) − \sqrt{Lr}]})^{2}\}

(4.7)

where Ψ(L) = Ψ_{λR_{λ}}(L).

Proof. μ being an invariant measure of P or (P_{t}), 0 is a spectral point of L with the largest real part.

At first we see why (4.3) is true and how the factor c(p) appears. Define similarly

λ_{0}^{ℂ}(p) := inf ; F ∈L_{0}^{p}(μ;ℂ) ⋂ 𝔻_{p}(L) \frac{∥LF ∥_{p}}{∥F ∥_{p}}

(the only difference from the definition of λ_{0}(p) is: here F is complex-valued). For any complex number λ such that 0 < |λ| < λ_{0}^{ℂ}(p), we have for any F ∈ L_{0}^{p}(μ; ℂ),

∥(λ − L)F ∥_{p} ≥ ∥LF ∥_{p} − |λ|∥F ∥_{p} ≥ (λ_{0}^{ℂ}(p) − |λ|)∥f∥_{p}.

Hence λ−L : L_{0}^{p}(μ; ℂ) → L_{0}^{p}(μ; ℂ) has a bounded inverse. But for any F ∈ L^{p}(μ; ℂ), F = (F − μ(F)) + μ(F) ∈ L_{0}^{p}(μ; ℂ) ⊕ ℂ and, restricted to ℂ, (λ − L)c = λc is also invertible. Hence λ − L : L^{p}(μ; ℂ) → L^{p}(μ; ℂ) has a bounded inverse. This means that λ belongs to the resolvent set of L for any complex λ with 0 < |λ| < λ_{0}(p), proving so λ_{gap}(p) ≥ λ_{0}^{ℂ}(p).

We now control λ_{0}^{ℂ}(L) by λ_{0}(p). The trick is to introduce two independent Rademacher random variables ξ_{1}, ξ_{2} defined on (Ω, F, ℙ) (i.e., ℙ(ξ_{k} = ±1) = 1/2), to apply the Khinchine inequality: for some universal constants c_{1}(p), c_{2}(p) depending only on p with c_{1}(2) = c_{2}(2) = 1,

\frac{1}{c_{2}(p)}\sqrt{a^{2} + b^{2}} ≤ ∥aξ_{1} + bξ_{2}∥_{p} ≤ c_{1}(p)\sqrt{a^{2} + b^{2}}, ∀a, b ∈ ℝ.

Now the idea goes as follows: for any F = f + ig ∈ L_{0}^{p}(μ; ℂ) where f, g ∈ L_{0}^{p}, we have on (E × Ω, μ × ℙ) (below 𝔼^{ω} denotes the expectation w.r.t. ω),

∥LF ∥_{p}^{p} = ∫_{E}[(Lf)^{2} + (Lg)^{2}]^{p/2}dμ

≥ \frac{1}{(c_{1}(p))^{p}} 𝔼^{ω} ∫_{E} |L(ξ_{1}f + ξ_{2}g)|^{p}dμ

≥ (\frac{λ_{0}(p)}{c_{1}(p)})^{p} 𝔼^{ω} ∫_{E} |ξ_{1}f + ξ_{2}g|^{p}dμ

≥ (\frac{λ_{0}(p)}{c_{1}(p) · c_{2}(p)})^{p} ∫_{E} |f^{2} + g^{2}|^{p/2}dμ

= (\frac{λ_{0}(p)}{c_{1}(p) · c_{2}(p)})^{p} ∥F ∥_{p}^{p}.

Thus λ_{gap}(p) ≥ λ_{0}^{ℂ}(p) ≥ c(p)λ_{0}(p) with c(p) = [c_{1}(p)c_{2}(p)]^{−1}, proving so (4.3).

We now turn to (4.4). For any f ∈ L_{0}^{p}(μ) with ∥f∥_{p} = 1, denoting t = ∥Lf∥_{p}, we have by (3.6) in the proof of Lemma 3.4 that

\begin{aligned}1 = ∥f∥_{p} & ≤ Ψ_{λR_{λ}}(L) · \frac{1}{λ}∥(λ − L)f∥_{p} + (\frac{L}{λ}∥(λ − L)f∥_{p})^{1/p′} · ∥f∥_{1}^{1/p} \\ & ≤ Ψ_{λR_{λ}}(L) + \frac{Ψ_{λR_{λ}}(L)}{λ}t + (\frac{L}{λ}(λ + t))^{1/p′} · ∥f∥_{1}^{1/p} \end{aligned}

But by Proposition 2.6 and 2.9,

χ^{(R)} ((\frac{∥f∥_{1}}{4} )^{p′}) ∥f∥_{1} ≤ ∥Lf∥_{1} ≤ ∥Lf∥_{p} = t.

Then ∥f∥_{1} ≤ κ_{p}(t) given in (4.5). Substituting it into the previous inequality we see that t = ∥Lf∥_{p} satisfies the inequation (4.4), so does t = λ_{0}(p) by the right continuity of the l.h.s. of (4.4) in t ≥ 0.

Finally for (4.7) in the case p = 2, notice that for any λ, L > 0, r ∈ (0, 1) verifying (4.6), t = λ_{0}(2) satisfies

• either κ_{2}(t) > r under which rχ^{(R)} (\frac{r^{2}}{16}) ≤ t;

• or κ_{2}(t) ≤ r, under which the inequation (4.4) becomes

1 − Ψ_{λR_{λ}}(L) ≤ \frac{Ψ_{λR_{λ}}(L)}{λ}t + (\frac{L}{λ}(λ + t)r)^{1/2} .

Solving this inequation, we obtain the first inequality in (4.7). The second is elementary.

We prepare the following:

Lemma 4.2. Let P be a Markov operator with μ = μP. Then for any λ, L > 0 and 0 < r < 1,

inf\{|λ|; 0 ≠ λ ∈ Σ_{L_{ℂ}^{2}}(I − P)\} ≥ [r · χ_{P} (\frac{r^{2}}{16})] ∧ [1 − Ψ_{P} (L) − \sqrt{Lr}] . (4.8)

Proof. The idea is same as in Theorem 4.1. We begin with the formula precedent to (3.6) with π = P: for any f ∈ L_{0}^{2}(μ),

∥P f∥_{2} ≤ Ψ_{P} (L) · ∥f∥_{2} + \sqrt{L∥f∥_{2} · ∥P f∥_{1}} ≤ Ψ_{P} (L) · ∥f∥_{2} + \sqrt{L∥f∥_{2} · ∥f∥_{1}}

where it follows

∥(I − P)f∥_{2} ≥ (1 − Ψ_{P} (L))∥f∥_{2} − \sqrt{L∥f∥_{2} · ∥f∥_{1}}.

The inequality (2.16) in Proposition 2.6 together with (2.18) yields

χ_{P} (\frac{∥f∥_{1}^{2}}{16∥f∥_{2}^{2}} ) · ∥f∥_{1} ≤ ∥(I − P)f∥_{1} ≤ ∥(I − P)f∥_{2}.

Substituting this control on ∥f∥_{1} into the previous inequality and letting κ(u) := 1 ∧ inf\{r ∈ (0, 1); r · χ^{(R)} (\frac{r^{2}}{16}) > u\}, we obtain

∥(I − P)f∥_{2} ≥ (1 − Ψ_{P} (L))∥f∥_{2} − \sqrt{L · κ (\frac{∥(I − P)f∥_{2}}{∥f∥_{2}} )} · ∥f∥_{2}.

Hence

λ_{0} := λ_{0}(2) = inf_{0≠f∈L_{0}^{2}(μ)} \frac{∥(I − P)f∥_{2}}{∥f∥_{2}}

verifies

1 − Ψ_{P} (L) ≤ λ_{0} + \sqrt{L · κ(λ_{0})}.

Now there are only two possibilities:

• either κ(λ_{0}) > r: in such case λ_{0} ≥ r · χ^{(R)} (\frac{r^{2}}{16});

• or κ(λ_{0}) ≤ r: then λ_{0} ≥ 1 − Ψ_{P} (L) − \sqrt{Lr}.

The desired estimate follows by (4.3).

The following estimate is perhaps more explicit:

Theorem 4.3. Assume the same as in Theorem 4.1 (for P or (P_{t})). Let

g(L, r, λ) := [r · χ_{λR_{λ}} (\frac{r^{2}}{16})] ∧ [1 − Ψ_{λR_{λ}}(L/L^{2}(μ)) − \sqrt{Lr}] .  (4.9)

Then for any λ, L > 0, r ∈ (0, 1) such that g(L, r, λ) > 0,

λ_{gap}(2) ≥ \frac{λ · g(L, r, λ)}{λ + g(L, r, λ)}.  (4.10)

Proof. Applying (4.8) to λR_{λ}, we get that the gap of λR_{λ} − I near 0 is greater that g(L, r, λ). Now (4.10) follows for Σ(λR_{λ} − I) = {λ/(λ − z) − 1; z ∈ Σ(L)} (by the spectral mapping theorem).

Remarks (4.i). The estimates (4.7) and (4.10) give explicit lower bounds of the spectral gap λ_{gap}(2) in terms of two quantities: the RUPI coefficient χ^{(R)}(r) and the tail norm parameter Ψ_{λR_{λ}}(L), refining Theorem 1.2. It generalizes the previous work of Aida [1] which is realized for continuous time symmetric hyperbounded Markov semigroups. Indeed in that case, we have the defected log-Sobolev inequality below (by Gross [19])

⟨f^{2} log f^{2}⟩ ≤ ⟨f^{2}⟩ log⟨f^{2}⟩ + aE[f] + b⟨f, f⟩, ∀f ∈ 𝔻(E).

Applying it to f = λR_{λ}g =: πg where 0 ≤ g ∈ B_{2}^{+}(1), and noting that E[λR_{λ}g] ≤ λ/4 (by spectral decomposition and \sqrt{xy} ≤ (x + y)/2), we obtain for all L > 1

∥1_{[πg>L]}πg∥_{2}^{2} ≤ \frac{1}{log L^{2}} [⟨(πg)^{2} log(πg)^{2}⟩ + e^{−1}] ≤ \frac{1}{log L^{2}} [\frac{aλ}{4} + b + \frac{1}{e}]

Thus Ψ_{λR_{λ}}(L) is smaller than the square root of the last quantity. Substituting it into (4.7) and (4.10), we get similar estimates as those in [1].

Remarks (4.ii). The estimates (4.7) and (4.10) are quite far from being sharp in concrete situations. By the abundant and remarkable works on sharp estimates of the spectral gap of the Laplacian on Riemannian manifolds (see [5], [6], [4] etc), the author is convinced that the sharp estimate is a mixing of many factors: the RUPI and the tail norm are only two rough parameters (coming only from measure). Nevertheless the estimates (4.7), (4.10) are universal (by the necessity in Theorem 1.2) and are useful for some infinite dimensional models.

Remarks (4.iii). The careful reader see that the RUPI is not assumed in the two results above: the spectral gap in L^{2} holds (then RUPI is true) if (4.6) holds for some λ, L > 0, r ∈ (0, 1). This is quite strange: it is possible that, e.g., for each N ≥ 1, you can find some non-RUPI Markov semigroup (even a finite Markov matrix) such that χ^{(R)}(1/16 × 2^{4N}) > 0. In that case Theorem 4.1 implies that 1 ≤ Ψ_{λR_{λ}}(L) + \sqrt{L}/2^{N} for all λ, L > 0. Strange, no?

4.3 Estimate of rate of exponential convergence

Hino characterizes the exponential convergence (1.4) of P_{t} to its equilibrium measure μ in terms of conditions (E) and (I) (see Theorem 1.1 in the Introduction).

For the continuous time symmetric semigroup (P_{t}), δ = λ_{gap}(2) = λ_{0}(2) and C(δ) = 1 in (1.4). But in the nonsymmetric case, whether is possible (and how) to control δ and C(δ) in terms of λ_{gap}(2)?

The general answer is unfortunately: NO. This can be seen for the simple Example 4(uniform motion on the circle, continued). Its resolvents R_{λ} for λ > 0 are compact on L^{2}(S, dx), but (1.4) is obviously false. The reason is: the spectrum of its generator ∂_{t} in L^{2}(S, dx) is {in; n ∈ ℤ}, completely located on the imaginary axe iℝ.

This example shows that other parameters than χ^{(R)}(r), Ψ_{λR_{λ}}(L) should be used for estimating the exponential convergence rate δ and the associated constant C(δ) in (1.4). This is done in the following

Proposition 4.4. Given a Markov semigroup (P_{t})_{t∈𝕋} (𝕋 = ℕ or ℝ^{+}) with invariant measure μ, let V_{t} = P_{t}^{∗}P_{t}. Assume that there exists some L, r, T = T (L, r) > 0 such that

λ(T, L, r) := [r · χ_{V_{T}} (\frac{r^{2}}{16})] ∧ [1 − Ψ_{V_{T}} (L/L^{2}(μ)) − \sqrt{Lr}] > 0.  (4.11)

Then the exponential convergence (1.4) holds in L^{2}(μ) with

δ = \frac{1}{2T} | log(1 − λ(T, L, r))|, C(δ) ≤ \sqrt{\frac{1}{1 − λ(T, L, r)}}.  (4.12)

Proof. Write V = V_{T}. By Lemma 4.2,

λ_{0} := inf_{0≠f∈L_{0}^{2}(μ)} \frac{∥(I − V_{T})f∥_{2}}{∥f∥_{2}} ≥ λ(T, L, r).  (4.13)

Thus for any f ∈ L_{0}^{2}(μ),

⟨P_{T} f, P_{T} f⟩_{μ} = ⟨f, V_{T} f⟩_{μ} ≤ (1 − λ_{0})⟨f, f⟩_{μ}.

Consequently for any t = nT + r with n ∈ ℕ and r ∈ [0, T), we have

∥P_{nT +r}f∥_{2} ≤ (1 − λ_{0})^{n/2}∥P_{r}f∥_{2} ≤ exp (−\frac{1}{2T} | log(1 − λ_{0})| · t) · \sqrt{\frac{1}{1 − λ_{0}}} · ∥f∥_{2}

where follows the desired estimate (1.4) with δ, C(δ) given by (4.12).

Remarks (4.iv): The condition (4.11) is just the combination of the condition (E) and the condition (I) in Theorem 1.1 (easy from Hino [22]) for characterizing the exponential convergence (1.4). Hence this proposition can be regarded as a quantitative version of his result.

5 Absolute continuous kernels: weak compactness and essential spectral gap

5.1 A weak compactness criterion

From now on we present several very general situations where the RUPI is satisfied. We begin with

Proposition 5.1. Given a semigroup (π_{t})_{t∈𝕋} of positive operators on L^{p}(μ) where 𝕋 = ℕ or ℝ^{+}. Assume the absolute continuity condition, i.e., for each 0 < t ∈ 𝕋, π̃_{t}(x, dy) = p_{t}(x, y)μ(dy) for some kernel realization π̃_{t}(x, dy) of π_{t}. If (π_{t}) is μessentially irreducible (i.e., for any A ∈ B charged by μ, R_{λ}1_{A} > 0, μ − a.e.), then (π_{t}) is RUPI .

Proof. Fix λ > 0 sufficiently large such that λR_{λ} (with R_{λ} defined at the beginning of Section 2) is bounded and positive improving (by the essential irreducibility). It can be realized by an absolute continuous kernel by the proof of Lemma 2.3. Now by [40](p???), λR_{λ} allows a kernel realization which is irreducible. By Proposition 2.4, λR_{λ} is RUPI. That is equivalent to the UPI of λR_{λ} (which is the RUPI of (π_{t})), for λR_{λ} and its resolvent have the kernel realizations which are mutually absolute continuous.

Remarks (5.i). By Lemma 2.1, all elliptic or hypoelliptic diffusions on Riemannian manifolds are UPI w.r.t. any reference measure ν equivalent to the Riemannian volume measure dx.

By this proposition, if E is countable and μ charges each point of E, then any irreducible (infinite) positive matrix (P (x, y)), regarded as a kernel, is RUPI.

In [53] or [55] (Theorem 2.3), we proved that for an absolutely continuous signed kernel π, if it is uniformly integrable in L^{p}(μ) (i.e., ∥π∥_{tail} = 0), then π^{2} is compact.

That result suggests that under the absolute continuity assumption, some stronger conclusion than the WSGPs could be drawn out. The following result, going in this direction, extends the classical Rellich-Kondrachov compactness theorem in the theory of Sobolev spaces. It might be useful in quite wide situation.

Theorem 5.2. Let 1 < p < +∞, ρ a σ-finite nonnegative measure (maybe finite) on (E, B) and L^{0}(ρ) is the space of all real measurable functions equipped with the topology of convergence in measure ρ.

(a) Assume that π : L^{p}(ρ) → L^{p}(ρ) is a bounded operator given by the absolute continuous kernel p(x, y)ρ(dy). If

∫ |p(x, ·)|ρ(dx) ∈ L^{p′}(ρ),  (5.1)

then π is compact from L^{p}(ρ) to L^{1}(ρ).

(b) Assume that π : L^{p}(ρ) → L^{p}(ρ) is a bounded positive operator given by the absolute continuous kernel p(x, y)ρ(dy). Then π is compact from L^{p}(ρ) to L^{0}(ρ).

(c) Let (π_{t})_{t≥0} be a semigroup of bounded positive operators on L^{p}(E, ρ) with generator L and,

ω_{0} := lim_{t→∞} \frac{1}{t} log ∥π_{t}∥_{L^{p}(ρ)→L^{p}(ρ)}.

Assume that π_{t} is given by some absolute continuous kernel p_{t}(x, y)ρ(dy) (∀t > 0). Then for any λ > ω_{0} and for any r, t > 0,

(λ − L)^{−r/2} := \frac{1}{Γ (\frac{r}{2})} ∫_{0}^{∞} t^{\frac{r}{2} −1}e^{−λt}π_{t}dt

and π_{t} are compact from L^{p}(ρ) to L^{0}(ρ). In particular, for any sequence (f_{n}) such that sup_{n} ∥(λ − L)^{r/2}f_{n}∥_{p} < +∞, there exists some subsequence (f_{n_{k}}) which converges in L^{0}(ρ).

(d) In part (c), if moreover p = 2 and (π_{t}) is symmetric on L^{2}(ρ), then for any sequence (f_{n}) such that sup_{n}⟨f_{n}, (1 + ω_{0} − L)f_{n}⟩_{ρ} < +∞, there exists some subsequence (f_{n_{k}}) which converges in measure ρ.

Below B_{q,ρ}(1) denotes the unit ball in L^{q}(ρ) (q ∈ [1, +∞]). All those weak compactness results are based on

Lemma 5.3. Let N(x, dy) = p(x, y)ρ(dy) where ∫ |p(x, y)|ρ(dy) < +∞ for ρ − a.e. x. Then

{Kf; ∥f∥_{∞} ≤ 1}

is relatively compact in L^{0}(ρ).

Proof. This lemma is established in the proof of [55], Theorem 2.3 (by means of Bourbaki’s theorem). We reproduce here that argument because of its simplicity.

Without loss of generality we can assume that ∫ |p(x, y)|ρ(dy) < +∞ for all x ∈ E. It is well known thatB_{∞,ρ}(1) equipped with the weak^{∗} topology σ(L^{∞}, L^{1}(ρ)) is compact. By Bourbaki’s theorem, the unit ball B_{∞,ρ}(1) in L^{∞}(ρ) equipped with the weak^{∗} topology σ(L^{∞}, L^{1}(ρ)) is also metrisable, then sequentially compact, i.e., for any sequence (f_{n}) contained in the unit ball B_{∞,ρ}(1), there exists some subsequence (f_{n_{k}}) which converges to some f ∈ B_{∞,ρ}(1) w.r.t. σ(L^{∞}, L^{1}(ρ)).

Now for every x ∈ E, p(x, ·) ∈ L^{1}(ρ), then Kf_{n_{k}}(x) = ⟨p(x, ·), f_{n_{k}}(·)⟩_{ρ} → ⟨p(x, ·), f(·)⟩_{ρ}. Thus Kf_{n_{k}} converges in measure ρ to Kf, the desired claim.

Proof of Theorem 5.2. (a): The dual π^{∗} on L^{p′}(ρ) can be realized as

f → π^{∗}f(·) = ∫_{E} f(x)p(x, ·)ρ(dx), ∀f ∈ L^{∞}(ρ),

Thus by Lemma 5.3, π^{∗}(B_{∞,ρ}(1)) is relatively compact in L^{0}(ρ) (then sequentially pre-compact for L^{0}(ρ) is metrisable), In further for any f ∈ L^{∞}(ρ) with ∥f∥_{∞} ≤ 1,

|π^{∗}f(·)| ≤ ∫_{E} |p(x, ·)|ρ(dx) ∈ L^{p′}(ρ)

Thus by Lebesgue’s dominated convergence theorem, π^{∗}(B_{∞,ρ}(1)) is sequentially precompact in L^{p′}(ρ), i.e., π^{∗} : L^{∞} → L^{p′}(ρ) is compact. But by Schauder’s theorem, π : L^{p}(ρ) → L^{1}(ρ) is compact iff π^{∗} : L^{∞} → L^{p′}(ρ) is compact.

(b): We take some element ϕ > 0, a.e in L^{p}(ρ) with ρ(ϕ^{p}) = 1 and set ν = ϕ^{p}ρ. Consider the isomorphism I_{ϕ} : f → fϕ from L^{p}(ν) to L^{p}(ρ). The associated transformed operator

S := I_{ϕ}^{−1}πI_{ϕ}

is a bounded and positive operator on L^{p}(ν) given by the absolute continuous kernel \frac{ϕ(y)}{ϕ(x)}p(x, y)ν(dy). S on L^{p}(ν) satisfies condition (5.1) which means simply S_{ν}^{∗}1 ∈ L^{p′}(ν) (this last property is true for 1 ∈ L^{p′}(ν)). Applying (a) to S on L^{p}(ν), we see that S : L^{p}(ν) → L^{1}(ν) is compact. Therefore π(B_{p,ρ}(1)) = I_{ϕ}S(B_{p,ν}(1)) is relatively compact in L^{0}(ρ) (since ρ, ν are equivalent), where B_{p,ν}(1) is the unit ball in L^{p}(ν), the desired result.

(c): The compactness of π_{t}, (λ−L)^{−r/2} (for all t, r > 0 and λ > ω_{0}) from L^{p}(ρ) to L^{0}(ρ) follows by part (b). The last claim in this part is simply a more transparent (and useful) version of the compactness of (λ − L)^{−r/2} from L^{p} to L^{0}.

(d): This is a particular case of the last claim in part (c), with r = 1 and λ = ω_{0} + 1. □

Remarks (5.ii). The author has communicated Part (d) to J. Picard [38] on 1999, which is useful for his investigation on harmonic mappings.

Corollary 5.4. (due to de Pagter and Schepp [7]) If π is an absolute continuous nonnegative kernel, bounded on L^{p}(μ) (μ is a probability measure), then ∥π∥_{tail} equals to the measure of non-compactness β(π) of Nussbaum, defined by

β(π) := inf{δ > 0; ∃(f_{k})_{1≤k≤n} ⊂ L^{p} : π(B_{p}(1)) ⊂ ⋃_{1≤k≤n} B(f_{k}, δ)}
= inf{δ > 0; ∃ compact K ⊂ L^{p} : dist(π(B_{p}(1)); K) < δ}

where B(f_{k}, δ) is the ball (in L^{p}) of radius δ centered at f, and

dist(A, B) := max {sup_{f∈A} inf_{g∈B} ∥f − g∥_{p}, sup_{g∈B} inf_{f∈A} ∥f − g∥_{p}}

is the distance between A and B.

Proof. It is easy to see that ∥π∥_{tail} ≥ β(π) (even without the absolute continuity assumption). To establish the inverse, consider the set A_{L} := {f_{L} := (f ∧ L) ∨ (−L); f ∈ π(B_{p}(1))}. It is relatively compact in L^{0}(μ) by Theorem 5.2, so in L^{p}(μ). But the distance between A_{L} and π(B_{p}(1)) does not exceed Ψ_{π}(L/p, μ). Thus β(π) ≤ Ψ_{π}(L). Letting L go to infinity, we obtain the desired inverse inequality.

5.2 Estimates of the essential spectral gap by means of tail norm and weak Poincaré inequality

We now apply Theorem 1.4 and the results of Section 3 to the continuous time case.

Corollary 5.5. Let 𝕋 = ℝ^{+} and (P_{t}) is a C_{0}-semigroup of positive operators on L^{p}(μ) satisfying (A2) (in Lemma 3.4) and P_{t}, t > 0 are realized as absolute continuous kernels absolute continuous w.r.t. ν.

(a) If there are r_{0} ∈ ℝ^{+} and a non-increasing function β(r) on (r_{0}, +∞) such that

∥f∥_{p} ≤ r∥Lf∥_{p} + β(r)∥f∥_{1}, ∀f ∈ 𝔻_{p}(L), ∀r > r_{0},

then

λ_{ess−gap}(−L) := inf\{|λ|; λ ∈ Σ_{ess}(−L)\} ≥ \frac{1}{r_{0}} .  (5.2)

(b) If there are r_{0} ∈ ℝ^{+} and a non-increasing function β(r) on (r_{0}, +∞) such that

⟨f^{p}⟩_{μ} ≤ r⟨−Lf, f^{p−1}⟩_{μ} + β(r)⟨f⟩_{μ}^{p}, ∀0 ≤ f ∈ 𝔻_{p}(L), r > r_{0}  (5.3)

then the essential spectral radius r_{ess}(P_{t}) of P_{t} is smaller than exp(−t/r_{0}), and

inf\{Reλ; λ ∈ Σ_{ess}(−L)\} ≥ \frac{1}{r_{0}} .  (5.4)

In particular, for any ε > 0, there exist some constant C ≥ 1 and a continuous projection J with finite dimensional range such that JP_{t} = P_{t}J (for all t ≥ 0), and

∥P_{t}(I − J)∥_{p} ≤ C · exp (−\frac{1}{r_{0} + ε}t) .  (5.5)

Proof. (a) By Theorem 3.5, lim_{δ→0+} ∥R_{δ}∥_{tail} ≤ r_{0}. Hence for any λ ∈ Σ_{ess}(−L), by the spectral mapping theorem, (δ + λ)^{−1} ∈ Σ_{ess}(R_{λ}). As R_{λ} can be realized as an absolute continuous kernel /μ, we obtain by (1.7) in Theorem 1.4,

lim sup_{δ→0+} \frac{1}{|δ − λ|} ≤ lim sup_{δ→0+} ∥R_{δ}∥_{tail} ≤ r_{0}

where (5.2) follows.

(b) By Proposition 3.7, ∥P_{t}∥_{tail} ≤ exp(−t/r_{0}). Hence by (1.7) in Theorem 1.4, r_{ess}(P_{t}) ≤ exp(−t/r_{0}). Recall the semi-spectral mapping theorem ([?]),

{e^{λt}; λ ∈ Σ_{ess}(L)} ⊂ Σ_{ess}(P_{t})

(the equality is unfortunately false in general!). Now (5.4) follows. It remains to establish (5.5).

As r_{ess}(P_{1}) ≤ exp(−1/r_{0}), then for any ε > 0, there exist some constant C_{1} ≥ 1 and a finite dimensional eigen-projection J of P_{1} such that

∥P_{n}(1 − J)∥_{p} = ∥[P_{1}(I − J)]^{n}∥_{p} ≤ C_{1} exp (−\frac{1}{r_{0} + ε}n) , ∀n ∈ ℕ.

The eigen-projection J, constructed by the Dunford-Taylor integral of the resolvent of P_{1}, commutes with P_{t} for each t ≥ 0. Now (5.5) follows from the above estimate with C := C_{1} · sup_{0≤t≤1} ∥P_{t}∥_{p}.

Theorem 5.6. Let (P_{t})_{t∈ℝ^{+}} be a semigroup of symmetric contractive positive operators on L^{2}(μ). Assume moreover that each P_{t} (t > 0) can be realized as an absolute continuous kernel w.r.t μ. Then the following properties are equivalent:

(1) the weak Poincaré inequality of operator type (3.4) holds;

(2) lim_{λ→0+} ∥R_{λ}∥_{tail} ≤ r_{0};

(3) the weak Poincaré inequality of form type (1.8) holds;

(4) ∥P_{t}∥_{tail} ≤ e^{−t/r_{0}} for all t ≥ 0;

(5) Σ_{ess}(−L) ⊂ [1/r_{0}, +∞);

This result allows a better understanding of Theorem 1.6 due to Wang.

Proof. By Theorem 3.5 and Proposition 3.7, we have (4) ⇒ (2) ⇔ (1) and (3) ⇔ (4).

(4) ⇒ (5) is contained in Theorem 1.5 due to Weis and de Pagter-Schep. (5) ⇒ (4) is easy. Indeed by the spectral decomposition and (5), P_{t} = B_{t} + K_{t}

where K_{t} is compact and ∥B_{t}∥_{2} ≤ exp(−t/r_{0}). Hence

∥P_{t}∥_{tail} = ∥B_{t}∥_{tail} ≤ ∥B_{t}∥_{2} ≤ exp(−t/r_{0}).

It remains to show that (2) ⇒ (5). This follows by Corollary 5.5(5.2) and the self-adjointness of L.

6 Applications: several examples

6.1 Generalized Ornstein-Uhlenbeck processes on an abstract Wiener space

Up till now the reader might have the impression that almost all positive essentially irreducible operators are RUPI . This impression is correct only in the finite dimensional case (by Proposition 5.1 and Remarks (5.iii)), but is wrong in the infinite dimensional situations, as we show now.

We begin with a fundamental example in Euclidean quantum fields: the generalized Ornstein-Uhlenbeck semigroup or semigroups of second quantification on an abstract Wiener space.

Let (T_{t} = e^{tA}) be a C_{0}-semigroup of symmetric contractions on some Hilbert space H (which represents the first quantization), such that KerA = {0}. Let W be some separable locally convex space equipped with the Gaussian measure μ such that (W, H, μ) constitutes a triplet of the so called abstract Wiener space. Let I_{n} : H^{⊗n} → C_{n} (the n-th chaos) be the Wiener-Ito-Gross multiple stochastic integral. It is well known that each F ∈ L^{2}(W, μ) admits the following chaos decomposition,

F = ⟨F ⟩_{μ} + ∑_{k=1}^{∞} I_{n}(f_{n}), f_{n} ∈ H^{⊗n}.  (6.1)

Now the second quantification of (T_{t}) is defined as

P_{t}F = ⟨F ⟩_{μ} + ∑_{k=1}^{∞} I_{n}(T_{t}^{⊗n}f_{n})  (6.2)

It is well known that (P_{t}) is a symmetric Markov semigroup, called generalized Ornstein-Uhlenbeck semigroup (corresponding to a continuous Gaussian Markov process). Its generator L, playing the role of the Hamiltonian in the free Euclidean quantum fields (see B. Simon [44]), is called generalized Ornstein-Uhlenbeck operator. When (T_{t} = e^{−t/2}), (P_{t}) is the standard Ornstein-Uhlenbeck semigroup, L is the standard Ornstein-Uhlenbeck operator used in the Malliavin Calculus.

By the spectral decomposition of (P_{t}) above, we see that

λ_{gap}(L) = inf{λ > 0; λ ∈ Σ_{L^{2}(W,μ)}(−L)} = inf{λ; λ ∈ Σ_{H}(−A)} =: λ_{0}(A). (6.3)

If λ_{0}(A) > 0, then P_{t} (t > 0) has spectral gap near 1 (then RUPI), is hypercontractive by the pioneering work of Nelson, and satisfies the logarithmic Sobolev inequality of Gross. We claim now that if λ_{0}(A) = 0, then (P_{t}) is not RUPI.

In fact if λ_{0} = 0, then we can find a sequence of elements (h_{n}) ⊂ H orthogonal each to other, such that |h_{n}|_{H} = 1, |Ah_{n}|_{H} ⟶ 0. Let f_{n} = I_{1}(h_{n}). We see that

∥f_{n}∥_{L^{2}(W,μ)} = |h_{n}|_{H} = 1, ∥Lf_{n}∥_{L^{2}(W,μ)} = |Ah_{n}|_{H} ⟶ 0.

But there exists no subsequence (f_{n_{k}}) which converges in L^{1}(W, μ), because the norms in L^{p}(W, μ), p ∈ [1, +∞) are all equivalent on C_{1}. Therefore (P_{t}) does not verify the WSGP (C4b) in Theorem 2.8, then it is not RUPI.

We turn now to the discussion of the RTNC (or equivalently condition (E) in the actual symmetric case) under λ_{0}(A) = 0 (otherwise the RTNC holds).

Notice at first for each h ∈ H with |h|_{H} = 1, I(h) := I_{1}(h) is of the standard Gaussian law N (0, 1), and then

m_{2n} := 𝔼^{μ}(I(h))^{2n} = 2^{n}\sqrt{2}Γ (n + \frac{1}{2})

where Γ(α) := ∫_{0}^{∞} t^{α−1}e^{−t}. Choose a sequence (h_{n} ∈ 𝔻(A), n ≥ 1) such that |h_{n}|_{H} = 1 and

n^{2} · m_{2(n−1)}⟨Ah_{n}, h_{n}⟩_{H} → 0

(possible by λ_{0}(A) = 0). Let E^{L} be the Dirichlet form associated with (P_{t}). We have (see [52])

\begin{aligned}E^{L}(I(h_{n})^{n}, I(h_{n})^{n}) & = n^{2} ∫_{E}[I(h_{n})]^{2(n−1)}⟨Ah_{n}, h_{n}⟩_{H} \\ & = n^{2} · m_{2(n−1)}⟨Ah_{n}, h_{n}⟩_{H} → 0. \end{aligned}

Thus E^{L}(F_{n}, F_{n}) → 0 for F_{n} := I(h_{n})^{n}/\sqrt{m_{2n}}. Note ∥F_{n}∥_{2} ≡ 1. Let’s show \{F_{n}^{2}; n\} is not uniformly integrable.

Fix some h ∈ H with |h|_{H} = 1. Then F_{n} has the same distribution as I(h)^{n}/\sqrt{m_{2n}} for each n ≥ 1. If in contrary {F_{n}^{2}; n} were uniformly integrable, so would be {G_{n}^{2} := I(h)^{2n}/m_{2n}; n ≥ 1}. But by the expression of m_{2n}, G_{n}^{2} → 0, μ − a.s.. Hence we would get 𝔼^{μ}G_{n}^{2} → 0, which is in contradiction with 𝔼^{μ}G_{n}^{2} ≡ 1. That contradiction shows that (P_{t}) does not satisfies the RTNC in L^{2}(μ) by Theorem 3.8.

Summarizing the previous discussion, we have proved

Proposition 6.1. For the generalized Ornstein-Uhlenbek semigroup (P_{t}), we have the equivalence between the RUPI, the RTNC in L^{2}(μ), the spectral gap and the logarithmic Sobolev inequality of Gross.

Example 7. In the above framework let H = L^{2}(ℝ^{n}, dx), A = −Δ + m^{2} and W is the space of temperate distributions. The spectrum of A = −Δ + m^{2} on H = L^{2}(ℝ^{n}, dx) is [m^{2}, +∞). Thus if the mass m is zero, the associated second quantification semigroup (P_{t}) is neither RUPI (but it is well ergodic) nor satisfies the RTNC.

We now anticipate the next example by the following remark on the argument above on RUPI. The whole previous analysis is done on the first chaos C_{1} which is linear and is based on the property that all norms ∥ · ∥_{p}, 1 ≤ p < +∞ restricted to C_{1} are equivalent. This is not surprise for the Gaussian measure. But for other measures, the reader certainly guess that there are surprises. That’s what we will show now.

6.2 Semigroups of second quantification on a Poisson space

Röckner and Wang [42] has studied thoroughly the UPI and their inequality (v) in Theorem 1.3 of the free Dirichlet form on the configuration space over a Riemannian manifold, and found a necessary and sufficient condition (Theorem 7.1 in [42]). Let us examine a more general model from the point of view of second quantification, with the Malliavin Calculus as tool.

Let ρ be a σ-finite measure without atom on a Polish space S equipped with the Borel σ-field S. The Poisson space over (S, ρ) is given by

Ω = {∑_{i} δ_{x_{i}} (at most countable) ; x_{i} ∈ S}

equipped with the probability measure ℙ on F := σ(ω → ω(A)|A ∈ S) such that the generic point ω is a Poisson point process with the intensity measure ρ.

Let I_{n} : H^{⊗n} = L^{2}(S^{n}, ρ^{n}) → C_{n} (the n-th chaos on the Poisson space) be the Ito’s multiple stochastic integral, which is a continuous extension of the usual Stietjes integral below

I_{n}(f) := ∫ ⋯ ∫_{(E^{n})^{∗}} f(x_{1}, ⋯, x_{n})ω̃(dx_{1}) ⋯ ω̃(dx_{n}), ∀f ∈ L^{1}(E^{n}, ρ^{n}) ⋂ L^{2}(E^{n}, ρ^{n})

where (E^{n})^{∗} is the set of those elements (x_{1}, ⋯, x_{n}) ∈ E^{n} such that x_{i} ≠ x_{j} for all different i, j, ω̃ := ω − ρ is the compensated Poisson point process. The following properties are well known: for all f ∈ L^{2}(S^{n}, ρ^{n}), g ∈ L^{2}(S^{m}, ρ^{m}), I_{n}(f) = I_{n}(symf) and

𝔼I_{n}(f) · I_{m}(g) = \{ \begin{matrix}n!⟨symf, g⟩_{L^{2}(S^{n},ρ^{n})}, & if n = m; \\ 0, & otherwise \end{matrix}  (6.4)

where symf(x_{1}, ·, x_{n}) = (1/n!) ∑_{σ} f(x_{σ(1)}, ·, x_{σ(n)}) with σ varying on all rearrangements of {1, 2, ·, n}, is the symmetrisation of f.

The chaos decomposition (6.1) still holds for F ∈ L^{2}(Ω, ℙ) on the actual Poisson space. Given some symmetric sub-Markov semigroup (T_{t})_{t≥0} with generator (A, 𝔻_{2}(A)) on H = L^{2}(S, ρ) such that ρT_{t} ≤ ρ, its second quantification semigroup (P_{t}) on L^{2}(Ω, ℙ) is determined by (see Surgailis [45] and Meyer [30])

P_{t}1 := 1, P_{t}I_{n}(f) := I_{n}(T_{t}^{⊗n}f), ∀f ∈ L^{2}(S^{n}, ρ^{n}), n ≥ 1.  (6.5)

(P_{t}) so defined gives rise to a symmetric Markov semigroup of contractions on L^{2}(Ω, ℙ). Its generator will be still denoted by L. Surgailis [45](1984) showed that (P_{t}) is Markov iff (T_{t}) is sub-Markov (recall that this last property is not required on the abstract Wiener space), and construct the associated process.

Let us denote denote the Dirichlet form associated with A (resp. L) on L^{2}(ρ) (resp. L^{2}(ℙ)) by E^{A} (resp. E^{L}).

As in the notation of Theorem 4.1, let λ_{gap}(L) be the spectral gap near 0 of L in L^{2}(ℙ), and

λ_{0}(A) := inf{λ; λ ∈ Σ_{L^{2}(ρ)}(−A)}.

By the definition (6.5), it is easy to see that

λ_{gap}(L) = λ_{0}(A).  (6.6)

If λ_{0}(A) > 0, L has spectral gap, and then (P_{t}) is RUPI/ℙ. But strikingly (P_{t}) is not hypercontractive by Surgailis [45], [46](1985) (see Ledoux [27] for updated developments).

Now assume that λ_{0}(A) = 0. Notice that Ker(L) = ℝ (or equivalently (P_{t}) is ergodic), if and only if the kernel KerA of A in L^{2}(S, ρ) is {0}. So if KerA ≠ {0}, (P_{t}) is not ergodic, then not RUPI.

It remains to study the last possibility: λ_{0}(A) = 0 and KerA = {0}. At first let us restrict to the first chaos and see what can be derived from the RUPI of (P_{t}).

As on the abstract Wiener space we use the argument by absurd. Assume that there exists a sequence (f_{n})_{n∈ℕ} satisfy ∥f_{n}∥_{L^{2}(ρ)} = 1, and E^{A}[f_{n}] := E^{A}(f_{n}, f_{n}) → 0 (keep in mind that the existence of such a sequence entails the non-RUPI of (P_{t}) on the abstract Wiener space). Let F_{n} = I_{1}(f_{n}). By definition (6.5),

∥F_{n}∥_{L^{2}(Ω,ℙ)} = 1, LF_{n} = −I_{1}(Af_{n}) → 0 in L^{2}(ℙ).

By the assumed RUPI of (P_{t}) and Theorem 2.8 (or Theorem 1.3), F_{n} → 0 in L^{1}(ℙ). But now the loss of the equivalence between the norms ∥ · ∥_{1}, ∥ · ∥_{2} on the first chaos of Poisson does not allow us to reach a contradiction. To that end, let us impose the following condition on (f_{n}):

sup_{n} ∥f_{n}∥_{L^{q}(ρ)} < +∞

where 2 < q ≤ +∞. Hence sup_{n} ∥F_{n}∥_{q} < +∞ (a good exercise about Poisson point processes). This, together with F_{n} → 0 in probability ℙ, implies that F_{n} → 0 in L^{2}(ℙ), which is the desired contradiction (with ∥F_{n}∥_{L^{2}(Ω,ℙ)} = 1).

In other words, the RUPI of (P_{t}) implies the non-existence of sequence (f_{n}) satisfying

∥f_{n}∥_{L^{2}(ρ)} = 1, sup_{n} ∥f_{n}∥_{L^{q}(ρ)} < +∞, E^{A}[f_{n}] → 0.

This last property is obviously equivalent to

λ_{0}(A, q, R) := inf \{\frac{E^{A}[f]}{⟨f, f⟩_{ρ}} ; 0 ≠ f ∈ 𝔻(E^{A}), ∥f∥_{L^{q}(ρ)}^{2} ≤ R\} > 0, ∀R > 0. (6.7)

But it is obvious that as q increases, this property becomes weaker. Then the weakest is reached at q = +∞.

The above elegant argument is borrowed from Röckner and Wang (though they take directly q = +∞).

What is striking is that the above ”weakest” necessary condition is also sufficient to the RUPI of (P_{t}), as claimed in

Proposition 6.2. If the second quantification semigroup (P_{t}) of (T_{t}) over the Poisson space (Ω, ℙ) is RUPI(/ℙ), then (6.7) holds for each 2 < q ≤ +∞.

Inversely if (6.7) holds for some 2 < q ≤ +∞, then for any r > 0,

V ar_{ℙ}(F) ≤ r𝔼 (∥D_{·}F ∥_{L^{q}(ρ)})^{2} + \frac{1}{λ_{0}(A, q, 1/r)}E^{L}[F].  (6.8)

Here D_{x}F (ω) = F (ω + δ_{x}) − F (ω) ((x, ω) ∈ S × Ω) is the difference operator on the Poisson space, E^{A} (resp. E^{L}) is the Dirichlet form associated with (T_{t}) on H = L^{2}(S, ρ) (resp. (P_{t}) on L^{2}(Ω, ℙ)). In particular, the second quantification semigroup (P_{t}) of (T_{t}) over the Poisson space (Ω, ℙ) is RUPI (/ℙ).

This result, when q = ∞, is due to Röckner and Wang [42] in the particular case where A = ∇^{∗}∇ with ∇ being the gradient operator on a Riemannian manifold equipped with the Riemannian volume measure ρ(dx) = dx, see Remarks (6.i) below for details. Our main purpose here is to furnish a new proof of this result based on the difference operator D, which simplifies the original proof of Röckner and Wang [42] and allows us to get this extension.

The difference operator D is the basis of a type of Malliavin calculus on the Poisson space developed by Nualard and Vives [33], J. Jicard [37] (there are many variants due to Bismut, Bichtler-Gravereaux-Jacod, Dermoune-Kree-Wu, Carlen-Pardoux, Privault etc, besides [33], [37]). We recall several simple facts of that calculus.

At first the law of (x, ω) → ω + δ_{x} under \frac{ρ(A∩dz)}{ρ(A)} ⊗ ℙ(dω) is absolutely continuous w.r.t. ℙ and the density is ω(A)/ρ(A), for any A ∈ S with 0 < ρ(A) < ∞. Then D : L^{0}(Ω, ℙ) ⟶ L^{0}(E×W, ρ⊗ℙ) is well defined. Moreover the following integration by parts formula holds:

∫_{Ω}⟨F (ω + δ_{x}), f(x)⟩_{ρ(dx)}dℙ = ∫_{Ω} F (ω)ω(f)dℙ  (6.9)

for all F, f ≥ 0 measurable. (See [54] for ’proof’).

Let 𝔻_{2}(D) := {F ∈ L^{2}(ℙ)|DF ∈ L^{2}(ρ ⊗ ℙ)}, and D_{2} = D|_{𝔻_{2}(D)}. Then by Nualard and Vives [33],

D_{x}I_{n}(f) = nI_{n−1}((symf)(x, ·)), ∀f ∈ L^{2}(S^{n}, ρ^{n})

and its dual D_{2}^{∗} : L^{2}(E×W, ρ⊗ℙ) → L^{2}(ℙ) coincides with the Skorokhod anticipative integral:

D_{2}^{∗}I_{n−1}(f(x, ·)) = I_{n}(f), ∀f ∈ L^{2}(S^{n}, ρ^{n}).

Moreover D_{2}^{∗}D_{2} is the standard Ornstein-Uhlenbeck operator (or number operator), i.e.,

D_{2}^{∗}D_{2}I_{n}(f) = nI_{n}(f).  (6.10)

(Or equivalently −D_{2}^{∗}D_{2} is the generator in L^{2}(ℙ) of the standard Ornstein-Uhlenbeck semigroup (P_{t}) associated with T_{t}f = e^{−t}f.)

The following simple lemma is crucial: it relates any second quantification semigroup with the difference operator D on the Poisson space.

Lemma 6.3. (a) For F = I_{n}(f^{⊗n}) where f ∈ 𝔻_{2}(A), then

LF = D_{2}^{∗}(A_{x}D_{x}F)  (6.11)

where A_{x} denotes the operator A acting on x.

(b) For each F ∈ 𝔻(E^{L}), for ℙ−a.s. ω ∈ Ω, the function x → D_{x}F (ω) belongs to the extended Dirichlet space 𝔻_{e}(E^{A}) (see [12], pp35-36 for the precise definition), and

E^{L}(F, F) = ∫_{Ω} E^{A}(D_{·}F (ω), D_{·}F (ω))dℙ(ω).  (6.12)

Proof. (a). Let us verify (6.11) for F = I_{n}(f^{⊗n}), where n ∈ ℕ^{∗}, f ∈ 𝔻_{2}(A). By the facts recalled above,

\begin{aligned}D_{2}^{∗}A_{x}D_{2}I_{n}(f^{⊗n}) & = D_{2}^{∗}A_{x}(nf(x)I_{n−1}(f^{⊗(n−1)})) = nI_{n}(f^{⊗(n−1)} ⊗ Af) \\ & = LI_{n}(f^{⊗n}). \end{aligned}

(b). By part (a), the equality (6.12) holds for all elements F of the linear space C spanned by I_{n}(f^{⊗n}) where n ∈ ℕ and f ∈ 𝔻_{2}(A). Since C is dense in L^{2}(ℙ) and stable by (P_{t}) (i.e., P_{t}(C) ⊂ C for all t ≥ 0), C is a core of L, or (equivalently) dense in 𝔻_{2}(L) w.r.t. the graph norm ∥F ∥_{2} + ∥LF ∥_{2}, by Lemma 3.5 in [59]. Hence C is a core for the Dirichlet form E^{L}.

Now for any F ∈ 𝔻(E^{L}), there exists a sequence F_{n} ∈ C such that

∥F_{n} − F ∥_{2} → 0, E^{L}[F_{n} − F_{m}] → 0,

as n, m → ∞. By (6.12), 𝔼^{ℙ}E^{A}[D_{·}(F_{n} − F_{m})] → 0. Then there exists some subsequence F_{n_{k}} so that for ℙ − a.s. ω ∈ Ω, F_{n_{k}}(ω) → F (ω) and

E^{A}(D_{·}(F_{n_{k}} − F_{n_{l}}))(ω) → 0, D_{x}F_{n_{k}}(ω) → D_{x}F (ω), ρ(dx) − a.e.

as k, l → ∞. That implies D_{·}F (ω) ∈ 𝔻_{e}(E^{A}) by [12], pp35-36. And (6.12) for such general F follows from that for F_{n}.

Proof of Proposition 6.2. The necessity is already proved. Let us show the sufficiency. Let us begin with another version (6.13) of the condition (6.7).

Notice that for all f ∈ L^{2}(S, ρ) ⋂ L^{q}(ρ),

· either ⟨f, f⟩_{ρ} < (1/R)∥f∥_{L^{q}(ρ)}^{2},

· or ∥f∥_{L^{q}(ρ)}^{2} ≤ R⟨f, f⟩_{ρ}, which implies by definition of λ_{0}(A, R, q),

⟨f, f⟩_{ρ} ≤ \frac{1}{λ_{0}(A, q, R)}E^{A}[f].

Consequently for all f ∈ L^{2}(S, ρ) ⋂ L^{q}(ρ),

∫_{S} f^{2}dρ ≤ \frac{1}{R}∥f∥_{L^{q}(ρ)}^{2} + \frac{1}{λ_{0}(A, q, R)}E^{A}[f].  (6.13)

(this is similar to Lemma 7.3 in [42]). By the Poincaré inequality related with the difference operator (by (6.10) or [56], Remarks 1.4, (1.9)),

V ar(F) ≤ 𝔼 ∫_{S}(D_{x}F)^{2}dρ(x), ∀F ∈ L^{2}(ℙ).

By (6.13) and Lemma 6.3, for all F ∈ C,

\begin{aligned}𝔼 ∫_{S}(D_{x}F)^{2}dρ(x) & ≤ \frac{1}{R}𝔼∥D_{·}F ∥_{L^{q}(S,ρ)}^{2} + \frac{1}{λ_{0}(A, q, R)}𝔼E^{A}[D_{x}F (ω)] \\ & = \frac{1}{R}𝔼∥D_{·}F ∥_{L^{q}(S,ρ)}^{2} + \frac{1}{λ_{0}(A, q, R)}E^{L}[F] \end{aligned}

where (6.8) follows. Note that (6.8) is stronger than the inequality (v) in Theorem 1.3. Thus (P_{t}) is RUPI . □

Remarks (6.i) Proposition 6.2 extends Theorem 7.1 in [42] by Röckner and Wang, but our framework is more general. To convince the reader let us place in the framework of [42]: S is a complete connected Riemannian manifold, ρ(dx) = p(x)dx is a σ-finite measure with smooth density p(x) w.r.t. the volume measure dx, (T_{t}) is determined by the Dirichlet form which is the closure of

E^{A}(f, f) = \frac{1}{2} ∫_{S} |∇f|^{2}dρ, ∀f ∈ C_{0}^{∞}(S)

where ∇f is the gradient operator on S, |∇f|^{2} = ∇f · ∇f with · being the Riemannian inner product on the vector bundle.

The Dirichlet form (E^{min}, 𝔻(E^{min}) considered by Röckner and Wang [42], written on the Poisson space, is the closure of

E(F, F) = \frac{1}{2}𝔼 ∑_{i,j=1}^{n} ∂_{i}Φ(⋯)∂_{j}Φ(⋯)ω(∇f_{i} · ∇f_{j}), ∀F (ω) = Φ(ω(f_{1}), ⋯, ω(f_{n}))

where f_{i} ∈ C_{0}^{∞}(S) (where the index _{0} means compact support), Φ ∈ C_{b}^{∞}(ℝ^{n}) (the space of C^{∞} functions on ℝ^{n} with bounded derivatives of any order). We denote the space of all F of the above form by FC_{b}^{∞}.

For F (ω) = Φ(ω(f_{1}), ⋯, ω(f_{n})) ∈ FC_{b}^{∞}, letting F_{i}(ω) = ∂_{i}Φ(ω(f_{1}), ⋯, ω(f_{n})), we have

2𝔼E^{A}(D_{x}F) = 𝔼 ∫_{S} |∇_{x}F (ω + δ_{x})|^{2}dρ(x)

= 𝔼 ∫_{S} |∑_{i=1}^{n} F_{i}(ω + δ_{x})∇_{x}f_{i}|^{2} dρ(x)

= 𝔼 ∫_{S} ∑_{i,j=1}^{n} F_{i}(ω + δ_{x})F_{j}(ω + δ_{x})∇f_{i} · ∇f_{j}dρ

= 𝔼 ∑_{i,j=1}^{n} F_{i}(ω)F_{j}(ω)ω(∇f_{i} · ∇f_{j})

where the last equality follows by the integration by part formula (6.9) (the previous expression is contained in [41]). Thus by Lemma 6.3, E^{min} = E^{L} restricted to FC_{b}^{∞}. And consequently their Theorem 7.1 is a particular case of Proposition 6.2.

See Röckner [41] for sufficient conditions for the coincidence E^{min} = E^{L}. For example, the last coincidence is valid for S = ℝ^{d}, A = −Δ.

Example 8. (due to [42]) Let S = ℝ^{d}, A = −Δ. Then (P_{t}) is not RUPI/ℙ, since λ_{0}(−Δ, +∞, R) = 0 for each R > 0.

See Röckner and Wang [42] for a difficult example for which λ_{0}(A) = 0 but λ_{0}(−Δ, +∞, R) > 0 for each R > 0 (then (P_{t})_{t>0} is UPI/ℙ by Proposition 6.2.)

Remarks (6.ii) The author proved in [54] (non-published) that the standard Ornstein-Uhlenbeck semigroup on the Poisson space is not uniformly integrable. Much progress on the uniform integrability of general second quantification semigroups (P_{t}) is made recently by Gong, Ma and Zhang [13] (2001).

It seems that if λ_{0}(A) = 0, (P_{t}) would not satisfy the RTNC. That is worth to be further investigated.

6.3 Semigroups of Girsanov

6.3.1 Main result

Let (Ω, (X_{t}), (F_{t}), (ℙ_{x})_{x∈E}) be a conservative Markov-Hunt process where the space of states is Polish. We assume that

(Co1) Its Markov semigroup (P_{t})_{t≥0} is symmetric and ergodic w.r.t. μ. And its trajectories are continuous or equivalently Ω = C(ℝ^{+}, E) (the space of all Evalued continuous paths indexed by ℝ^{+}). This is equivalent to the locality of the associated Dirichlet form (E, 𝔻(E)).

(Co2) The carré du champs operator Γ(f, g) for f, g ∈ 𝔻(E) exists (E(f, g) = ∫_{E} Γ(f, g)dμ);

Let (M_{t})_{t≥0} is an additive ℙ_{μ}-local martingale. Consider a perturbation of ℙ_{μ} by means of Girsanov’s formula

ℚ_{μ}|_{F_{t}} := exp (M_{t} − \frac{1}{2}⟨M⟩_{t}) · ℙ_{μ}|_{F_{t}},  (6.14)

Q_{t}f(x) := 𝔼^{ℙ_{μ}} [f(X_{t}) exp (M_{t} − \frac{1}{2}⟨M⟩_{t}) | X_{0} = x] .  (6.15)

where ⟨M⟩ is the continuous quadratic variational process of M. When

𝔼^{ℙ_{μ}} exp (\frac{1}{2}⟨M⟩_{t}) < +∞, ∀t ≥ 0  (6.16)

the Doléans-Dade exponential local martingale (e(M)_{t} := exp (M_{t} − \frac{1}{2}⟨M⟩_{t}) , ∀t ≥ 0) is a true martingale by Novikov’s criterion. And ℚ_{μ} given in (6.14) defines a new Markov process law with transition semigroup (Q_{t}). This is a very current and important approach for solving stochastic differential equations or for constructing new Markov processes.

The study of the existence of invariant measure (and its regularity) and of the existence of spectral gap was begun by Shigekawa [64] (1987), and followed by Vintchger [65](1989) the author [66](1993), Aida and Shigekawa [61](1994), Hino [62], Aida [1] (1998), the author [55] (2000), Gong and the author [16](2000) etc. The following result is a further refinement.

Theorem 6.4. Given p ∈ (1, +∞). Assume (Co1), (Co2) and (Co3) There are r_{0} ≥ 0 and a non-increasing function β(r) on (r_{0}, +∞) such that

⟨f^{2}⟩_{μ} ≤ rE(f, f) + β(r)⟨|f|⟩_{μ}^{2}, ∀r > r_{0}, ∀f ∈ 𝔻(E).  (6.17)

(Co4) there exists some B ∈ L_{+}^{1}(μ) such that ⟨M⟩_{t} = ∫_{0}^{t} B(X_{s})ds (this is equivalent to “M ≠ 0” and the square integrability of M w.r.t. ℙ_{μ} under (Co2));

(Co5) For some λ > p^{′2}/2,

\begin{aligned}Λ(λB) & := sup \{∫_{E} λBf^{2}dμ − E(f, f)| f ∈ 𝔻(E), ⟨f^{2}⟩ ≤ 1\} \\ & < \frac{2}{r_{0}} (\frac{\sqrt{2λ}}{p′} − 1) . \end{aligned}  (6.18)

Then

(a) the Girsanov semigroup (Q_{t}) is a Markov C_{0}-semigroup on L^{p}(μ), satisfying

∥Q_{t}∥_{L^{p}(μ)→L^{p}(μ)} ≤ exp (\frac{a(λ)}{p}t) , a(λ) := \sqrt{\frac{2}{λ}}Λ(λB);  (6.19)

∥Q_{t}∥_{tail(p,μ)} ≤ e^{−κt}, κ := \sqrt{\frac{2}{λ}} · \frac{2 (\frac{\sqrt{2λ}}{p′} − 1) − Λ(λB)r_{0}}{r_{0}p} .  (6.20)

And there exists some a.s positive ψ ∈ L^{p′}(μ) with ⟨ψ⟩_{μ} = 1 and \sqrt{ψ} ∈ 𝔻(E) such that α := ψμ is the unique invariant absolute continuous /μ probability measure of (Q_{t}).

In particular as t goes to infinity, the empirical measure L_{t} := (1/t) ∫_{0}^{t} δ_{X_{s}}ds of (X_{t}) converges to α in (M_{1}(E), τ) exponentially fast in probability ℚ_{ν} for any initial measure ν ≪ μ with dν/dμ ∈ ⋃_{ε>0} L^{1+ε}(μ).

If moreover P_{t}, t > 0 are absolute continuous w.r.t. μ, then

r_{ess}(Q_{t}|_{L^{p}(μ)}) ≤ exp (−κt) , ∀t ≥ 0.  (6.21)

(b) Assume moreover that (P_{t}) is RUPI/μ. Then there exist δ, C > 0 such that

∥Q_{t}f − α(f)∥_{L^{p}(μ)} ≤ Ce^{−δt}∥f∥_{L^{p}(μ)}, ∀f ∈ L^{p}(μ), t ≥ 0.  (6.22)

If moreover (Q_{t}) is symmetric w.r.t. α = ψμ, then

∥Q_{t}f − α(f)∥_{L^{2}(α)} ≤ e^{−δt}∥f∥_{L^{2}(α)}, ∀f ∈ L^{2}(α), t ≥ 0.  (6.23)

For concrete applications of that result, we have to verify condition (Co5). Remark that if r_{0} = 0 (i.e., (P_{t})_{t>0} is uniformly integrable by Theorem 1.5), condition (Co5) means just that Λ(λB) < +∞ for some λ > p^{′2}/2.

6.3.2 Several corollaries

Corollary 6.5. Let R_{λ} := ∫_{0}^{∞} e^{−λt}P_{t}dt be the resolvent. Say that a measurable function f is in the Kato class (denoted by “f ∈ K”), if lim_{λ→+∞} ∥R_{λ}|f|∥_{∞} = 0. Assume (Cok) for k = 1, 2, 3, 4.

If B ∈ K, then (Q_{t}) is a C_{0}-semigroup on L^{p}(μ) for all 1 < p < +∞. If moreover

∥B∥_{K} := ∥R_{1}|B|∥_{∞} < \frac{8}{(2 + r_{0})^{2}}  (6.24)

then (Co5) is true for all p sufficiently large (resp. for all 1 < p < +∞ if r_{0} = 0). In particular all conclusions of part (a) in Theorem 6.4 hold for p ≫ 1 (resp. for each p ∈ (1, +∞) if r_{0} = 0), so does the part (b) if (P_{t}) is RUPI.

Proof. By Kuwae, we have for any f ∈ 𝔻(E),

∫ Bf^{2}dμ ≤ ∥R_{λ}B∥_{∞}E(f, f) + λ∥R_{λ}B∥_{∞} ∫ f^{2}dμ.  (6.25)

Thus Λ(V/δ(λ)) ≤ λ where δ(λ) := ∥R_{λ}B∥_{∞} goes to zero as λ → +∞. Thus by Step 1 of the proof of Theorem 6.4 below, (Q_{t}) is a C_{0}-semigroup on L^{p}(μ) for all p ∈ (1, +∞). If r_{0} = 0, we see that (Co5) is satisfied for each p ∈ (1, +∞).

Letting λ = 1 in (6.25) we obtain

Λ(B/∥B∥_{K}) ≤ 1.

From it we see easily that (Co5) holds for sufficiently large p under condition (6.24). Hence (6.20) holds for p ≫ 1.

Corollary 6.6. ([55], [16]) Assume (Cok) for k = 1, 2, 4. Assume that (P_{t}) satisfies the defected logarithmic Sobolev inequality,

∫ f^{2} log f^{2}/⟨f^{2}⟩_{μ}dμ ≤ λ_{0}E(f, f) + c ∫ f^{2}dμ

which will be denoted by LSI(λ_{0}, c). Then (Co3) holds with r_{0} = 0. If there is some δ > 0 such that

∫_{E} exp (\frac{(1 + δ)λ_{0}}{2} B) dμ < +∞,  (6.26)

(Co5) holds for all p ≫ 1. In particular, if (P_{t}) is supercontractive, i.e., ∀1 < p < q < +∞, ∥P_{t}∥_{p,q} < +∞, and if ∫ e^{δB}dμ < +∞ for some δ > 0, then (Co5) holds for every p ∈ (1, +∞).

Proof. The LSI(λ_{0}, c) implies the hyperboundedness of (P_{t}), then its uniform integrability in L^{2}(μ). Thus (Co3) holds with r_{0} = 0 by Theorem 1.5. For any V ≥ 0, we get by LSI(λ_{0}, c)

Λ(V) ≤ sup \{∫ V f^{2} − \frac{1}{λ_{0}} ∫ f^{2} log f^{2}dμ| ⟨f^{2}⟩_{μ} = 1\} + \frac{c}{λ_{0}}

= \frac{1}{λ_{0}} (log ∫ e^{λ_{0}V} dμ + c)

where the second equality is Donsker-Varadhan’s variational formula for entropy. Hence (Co5) holds for p ≫ 1 once if (6.26) is verified. If (P_{t}) is supercontractive, by Gross’ theorem, for any λ_{0} > 0, the LSI(λ_{0}, c)

holds for some c > 0. Thus the last conclusion is true.

Example 9. Let E be a complete connected Riemannian manifold and V ∈ C^{∞}(E) such that μ(dx) := e^{−2V} dx is a probability measure. Assume that Ric + 2HessV ≥ −K for some constant K.

Consider the operator L = \frac{1}{2}Δ − ∇V where Δ, ∇ are respectively the Laplacian and the gradient on E. Let (X_{t}) be the diffusion with generator L (solution of the martingale problem), which is conservative and whose semigroup of transition (P_{t}) is symmetric w.r.t. μ.

Assume that ∫_{E} exp (λρ^{2}(x, o)) dμ(x) < +∞ for all λ > 0, where ρ(x, o) is the Riemannian distance and o ∈ E is fixed. Then by Wang [49], (P_{t}) is supercontractive. Obviously it satisfies (Co1), (Co2), and UPI.

Consequently for any measurable vector field b on E such that

∫_{E} e^{δ|b(x)|^{2}}dμ(x) < +∞

(|b(x)| being the Riemannian length), the Girsanov semigroup (Q_{t}) associated with the solution ℚ_{μ} by the Girsanov formula of the martingale problem corresponding to the generator L_{b} := L + b possesses all properties claimed in Theorem 6.4 for all 1 < p < +∞ (indeed B = |b|^{2}).

Example 10. Let (E, H, μ) be an abstract Wiener space and (P_{t}) is the standard Ornstein-Uhlenbeck semigroup. Its generator L is given by L = −\frac{1}{2}∇^{∗}∇ where ∇ is the Malliavin gradient. It satisfies (Co1), (Co2) and LSI(1, 0) by Gross’ theorem [19]. It is UPI.

Now given some measurable function b : E → H, consider the solution ℚ_{μ} by the Girsanov formula of generator L_{b} := L + b · ∇. Then B = |b|_{H}^{2}. If

∫ exp(λ|b|_{H}^{2})dμ < +∞ for some λ > 1/2  (6.27)

then the Girsanov semigroup (Q_{t}) has an ergodic invariant measure α = ψμ with 0 < \sqrt{ψ} ∈ L^{2+}(μ) ⋂ 𝔻(E), and has spectral gap in L^{p}(μ) for p ≫ 1.

If b = −∇V, then α is explicitly known as e^{−2V} dμ and (Q_{t}) is symmetric w.r.t. α. In that case condition (6.27) implies the spectral gap of (Q_{t}) in L^{2}(α). In such case, Hino [62] proved that spectral gap existence if (6.27) holds for some λ > 2.

Moreover condition (6.27) is sharp for the existence of spectral gap. Indeed let E = ℝ, μ = N (0, 1) and b(x) = (x^{2}/2 − (1 + x^{2})^{ε})′ where ε ∈ (0, 1/2). Then L_{b} = (1/2)d^{2}/dx^{2} − ((1 + x^{2})^{ε})′dx generates a symmetric Markov semigroup (Q_{t}) w.r.t. α = exp(−2(1 + x^{2})^{ε})dx/C (C being the normalized constant).

We see that such b verifies (6.27) for λ = 1/2 but not for λ > 1/2. However (Q_{t}) has no spectral gap in L^{2}(α) (otherwise by Ledoux [27], there are constants C, δ > 0 such that α({x; |x| > r}) ≤ Ce^{−δr} for all r > 0, which is obviously false by the explicit expression of α).

6.3.3 Proof of Theorem 6.4

We begin with several simple lemmas.

Lemma 6.7. For any measurable function V ≥ 0,

∥P_{t}^{V} ∥_{2} = exp (Λ(V)t)  (6.28)

where P_{t}^{V} f(x) := 𝔼^{ℙ_{x}}f(X_{t}) exp (∫_{0}^{t} V (X_{s})ds) (for f ≥ 0) is the Feynman-Kac semigroup and Λ(V) is defined as in (6.18).

Proof. If V is bounded, the generator of (P_{t}^{V}) being L + V, and (6.28) follows by the Rayleigh principle. In the general case, let V_{n} := V ∧ n, we have by monotone convergence,

∥P_{t}^{V} ∥_{2} = sup_{0≤f∈B_{2}(1)} ∥P_{t}^{V} f∥_{2}

= sup_{0≤f∈B_{2}(1)} sup_{n} ∥P_{t}^{V_{n}}f∥_{2} = sup_{n} ∥P_{t}^{V_{n}}∥_{2}

= sup_{n} exp [t sup {∫ V_{n}f^{2}dμ − E(f, f)| f ∈ 𝔻(E), ∥f∥_{2} ≤ 1}]

= exp [t sup {∫ V f^{2}dμ − E(f, f)| f ∈ 𝔻(E), ∥f∥_{2} ≤ 1}] = exp(tΛ(V)).

By the characterization of symmetric Hunt processes and their regular representation in Ma and Röckner [28], we can apply the theory of regular Dirichlet forms in [12] where the reader are referred for the terminologies below.

If the additive martingale M is in L^{2}(ℙ_{μ}), by (Co2), the Revuz measure associated with ⟨M⟩ is absolutely continuous w.r.t. μ. Hence B := dν_{⟨M⟩}/dμ ≥ 0 verifies ⟨B⟩_{μ} = 𝔼^{ℙ_{μ}}⟨M⟩_{1} = 𝔼^{ℙ_{μ}}M_{1}^{2} < +∞.

Definition 6.1. Assume (Co1), (Co2). For any f ∈ 𝔻(E), let M(f) be the martingale in the Fukushima decomposition of f̃(X_{t})−f̃(X_{0}) (f̃ being the quasi-continuous version of f) and let ν_{⟨M,M(f)⟩} be the Revuz measure associated with the continuous additive functional (⟨M, M(f)⟩_{t}), which is absolutely continuous w.r.t. μ. We define

∇_{M}f = \frac{dν_{⟨M,M(f)⟩}}{dμ} .  (6.29)

By [12], we can derive easily

Lemma 6.8. Assume (Co1), (Co2).

(a) ∇_{M(g)}f = 2Γ(f, g) and |∇_{M}f| ≤ \sqrt{2BΓ(f, f)}, ∀f, g ∈ 𝔻(E).

(b) for f, g ∈ 𝔻(E) ⋂ L^{∞}, ∇_{M}(fg) = g∇_{M}f + f∇_{M}g.

Lemma 6.9. Assume (Co1), (Co2), (Co4) and (6.16) for M. Let L_{M} be the extended generator of (Q_{t}) in L^{∞}(μ), i.e., f ∈ 𝔻_{∞}(L_{M}) iff there exists g ∈ L^{∞}(μ) such that Q_{t}f − f = ∫_{0}^{t} Q_{s}gds for all t ≥ 0, and L_{M}f := g.

If f ∈ 𝔻_{∞}(L_{M}), then f ∈ 𝔻(E) ⋂ 𝔻_{1}(L). Moreover Lf = L_{M}f − ∇_{M}f.

Proof. Fix f ∈ 𝔻_{∞}(L_{M}). Then (M_{t}^{Q}(f) := f(X_{t}) − f(X_{0}) − ∫_{0}^{t} L_{M}f(X_{s})ds, t ≥ 0) is a bounded ℚ_{μ}-martingale, which can be chosen (and then will assumed) to be continuous by (Co1) and the equivalence of ℙ_{μ} and ℚ_{μ} on F_{t} for all t ≥ 0. By Girsanov’s formula,

M_{t}(f) := M_{t}^{Q}(f) + ⟨M, M^{Q}(f)⟩_{t}

is a ℙ_{μ}-local martingale. We claim that 𝔼^{ℙ_{μ}}[M(f)]_{1} < +∞.

To that end, by the Burkholder-Davis-Gundy inequality, the optional quadratical variational process [M^{Q}(f)]_{t} = [M(f)]_{t} verifies

𝔼^{ℚ_{μ}}[M^{Q}(f)]_{1}^{q} ≤ c(p)𝔼^{ℚ_{μ}} sup_{t≤1} |M_{t}^{Q}(f)|^{2q} < +∞, ∀q > 1.

By Hölder’s inequality, we have

\begin{aligned}𝔼^{ℙ_{μ}}[M(f)]_{1} & = 𝔼^{ℚ_{μ}} exp (−M_{1} + \frac{1}{2}[M]_{1}) [M(f)]_{1} \\ & ≤ (𝔼^{ℚ_{μ}} exp (−q′M_{1} + \frac{q′}{2} [M]_{1}))^{1/q′} · ∥[M^{Q}(f)]_{1}∥_{L^{q}(ℚ_{μ})} \end{aligned}

and by Hölder’s inequality again, for any r ∈ (1, +∞)

𝔼^{ℚ_{μ}} exp (−q′M_{1} + \frac{q′}{2} [M]_{1}) = 𝔼^{ℙ_{μ}} exp (−(q′ − 1)M_{1} + \frac{q′ − 1}{2}[M]_{1})

≤ (𝔼^{ℙ_{μ}} exp (−r(q′ − 1)M_{1} − \frac{r^{2}(q′ − 1)^{2}}{2} [M]_{1}))^{1/r} (𝔼^{ℙ_{μ}} exp [\frac{(q′ − 1)}{2}(r + 1)r′[M]_{1}])^{1/r′}

≤ (𝔼^{ℙ_{μ}} exp [\frac{(q′ − 1)}{2}(r + 1)r′[M]_{1}])^{1/r′}

where the last inequality is true for exp (−r(q′ − 1)M_{t} − \frac{r^{2}(q′−1)^{2}}{2} [M]_{t}), being a local positive martingale, is a ℙ_{μ}-supermartingale. The last term above is finite by (6.16) once if (q′ − 1)(r + 1)r′ ≤ 1 (valid if q is sufficiently large). We have so proved 𝔼^{ℙ_{μ}}[M(f)]_{1} < +∞, the desired claim.

Thus f ∈ 𝔻(E) and M_{t}(f) defined above becomes f̃(X_{t}) − f̃(X_{0}) − ∫_{0}^{t}(L_{M}f − ∇_{M}f)(X_{s})ds. Since ∇_{M}f ∈ L^{1}(μ) by Lemma 6.8, we have by 𝔼^{ℙ_{μ}}(M_{t}(f)|F_{0}) = 0,

P_{t}f − f = ∫_{0}^{t} P_{s}(L_{M}f − ∇_{M}f)ds, ∀t ≥ 0

which yields f ∈ 𝔻_{1}(L) and Lf = L_{M}f − ∇_{M}f.

Proof of Theorem 6.4. Part (a). We divide its proof into five steps (only the first is crucial). Below we write E[f] := E(f, f), Γ[f] := Γ(f, f).

Step 1. First notice that Λ(λB) < +∞ for some λ > 1/2 implies (6.16) by Lemma 6.7. Thus (Q_{t}) is Markov. Fix f ∈ 𝔻_{∞}(L_{M}) such that f ≥ ε > 0. We have f ∈ 𝔻(E) by Lemma 6.9, then f^{θ} ∈ 𝔻(E) for any θ > 0. For any δ > 0 which will be chosen later, using 2ab ≤ δa^{2} + (1/δ)b^{2}, we have

|⟨∇_{M}f, f^{p−1}⟩_{μ}| ≤ ∫ \sqrt{2BΓ[f]}f^{p/2} · f^{p/2−1}dμ (by Lemma 6.8)

≤ δ ∫ B(f^{p/2})^{2}dμ + \frac{1}{2δ} ∫ Γ[f]f^{p−2}dμ

≤ \frac{δ}{λ} (Λ(λB) ∫ f^{p}dμ + E[f^{p/2}]) + \frac{1}{2δ} ∫ Γ[f]f^{p−2}dμ

where the last inequality follows from the definition of Λ(λB).

Since f ∈ 𝔻_{1}(L) ⋂ 𝔻(E) (by Lemma 6.9), we have by the derivation property of Γ,

∫ Γ[f]f^{p−2}dμ = \frac{4}{p^{2}} ∫ Γ(f^{p/2}, f^{p/2}) == \frac{4}{p^{2}} E[f^{p/2}]
⟨−Lf, f^{p−1}⟩_{μ} = ∫ Γ(f, f^{p−1})dμ = \frac{4(p − 1)}{p^{2}}E[f^{p/2}].

Thus by the previous control on ⟨∇_{M}f, f^{p−1}⟩_{μ}, we get

⟨−L_{M}f, f^{p−1}⟩_{μ} = ⟨−Lf, f^{p−1}⟩_{μ} + ⟨∇_{M}f, f^{p−1}⟩_{μ}

≥ (\frac{4(p − 1)}{p^{2}} − [\frac{δ}{λ} + \frac{1}{2δ} · \frac{4}{p^{2}} ]) E[f^{p/2}] − \frac{δ}{λ}Λ(λB) ∫ f^{p}dμ.

Choosing δ = \sqrt{2λ}/p the above inequality reads as

⟨−L_{M}f, f^{p−1}⟩_{μ} ≥ \frac{1}{p} [(\frac{4}{p′} − \sqrt{\frac{8}{λ}}) E[f^{p/2}] − \sqrt{\frac{2}{λ}}Λ(λB) ∫ f^{p}dμ]  (6.30)

Notice that the constant factor of E[f^{p/2}] above is positive for λ > p^{′2}/2. Hence for a(λ) := \sqrt{\frac{2}{λ}}Λ(λB),

⟨(a(λ)/p)f − L_{M}f, f^{p−1}⟩_{μ} ≥ 0, ∀f ∈ 𝔻_{∞}(L_{M}), f ≥ ε > 0.

Since ε ≤ Q_{t}f ∈ 𝔻_{∞}(L_{M}) and L_{M}Q_{t}f = Q_{t}L_{M}f for any t ≥ 0 (easy), and since s → Q_{s}h is continuous from ℝ^{+} to L^{1}(μ) for all h ∈ L^{∞} (directly by (6.15), the detail is left to the reader), we see that t → Q_{t}f = f + ∫_{0}^{t} Q_{s}L_{M}fds is continuously differentiable in L^{p}(μ), and

\frac{d}{dt}⟨(Q_{t}f)^{p}⟩_{μ} = p⟨L_{M}Q_{t}f, (Q_{t}f)^{p−1}⟩_{μ} ≤ a(λ)⟨(Q_{t}f)^{p}⟩_{μ}.

By Gronwall’s inequality, ⟨(Q_{t}f)^{p}⟩_{μ} ≤ exp(a(λ)t)⟨f^{p}⟩_{μ} (∀t ≥ 0) for all f ∈ 𝔻_{∞}(L_{M}) satisfying f ≥ ε for some constant ε > 0. We now extend it to all g ≥ ε > 0 in L^{∞}. Indeed for such g, we can verify by integration by parts

R_{n}^{Q}g := ∫_{0}^{∞} e^{−nt}Q_{t}gdt ∈ 𝔻_{∞}(L_{M}), L_{M}R_{n}^{Q}g = −g + nR_{n}^{Q}g.

Moreover by the continuity of t → Q_{t}g from ℝ^{+} to L^{0}(μ) (noted previously), g_{n} := nR_{n}^{Q}g → g, in measure μ. Thus e(M)_{t}g_{n}(X_{t}) → e(M)_{t}g(X_{t}) in measure ℙ_{μ}, then in L^{1}(ℙ_{μ}), for {g_{n}, n} is uniformly bounded by ∥g∥_{∞}. Thus Q_{t}g_{n} → Q_{t}g in L^{1}(μ), then in L^{p}(μ) (for |Q_{t}g_{n}| ≤ ∥g∥_{∞}).

Consequently the precedent estimate shown for ⟨(Q_{t}g_{n})^{p}⟩_{μ} becomes (as n → ∞)

⟨(Q_{t}g)^{p}⟩_{μ} ≤ exp(a(λ)t)⟨g^{p}⟩_{μ}, a(λ) := \sqrt{\frac{2}{λ}}Λ(λB)  (6.31)

for all ε ≤ g ∈ L^{∞}, and then for all 0 ≤ g ∈ L^{p}(μ). That yields (6.19), which entails that (Q_{t}) is a C_{0}-semigroup on L^{p}(μ) (under (Co1), (Co2), (Co4) and Λ(λB) < +∞ for some λ > p^{′2}/2).

Step 2. For all 0 < ε ≤ f ∈ 𝔻_{∞}(L_{M}), by (Co3) and (6.30), we have for any r > r_{0},

∫ f^{p}dμ ≤ rE[f^{p/2}] + β(r)[μ(f^{p/2})]^{2}

≤ \frac{rp}{b(λ, p)}⟨−L_{M}f, f^{p−1}⟩_{μ} + \frac{ra(λ)}{b(λ, p)} ∫ f^{p}dμ + β(r)[μ(f^{p/2})]^{2}

where b(λ, p) := \frac{4}{p′} − \sqrt{\frac{8}{λ}}. Then ∀0 < ε ≤ f ∈ 𝔻_{∞}(L_{M}) and for all r > r_{0} satisfying

ra(λ) < b(λ, p) (true for all r > r_{0} sufficiently close to r_{0} by (Co4))

∫ f^{p}dμ ≤ c(r, p, λ)⟨−L_{M}f, f^{p−1}⟩_{μ} + β_{2}(r)[μ(f^{p/2})]^{2},  (6.32)

where c(r, p, λ) := \frac{rp}{b(λ,p)−ra(λ)} and β_{2}(r) := β(r)/ (1 − \frac{ra(λ)}{b(λ,p)}).

The inequality (6.32) implies as in the proof of Proposition 3.7,

⟨(Q_{t}f)^{p}⟩_{μ} ≤ exp (−\frac{pt}{c(r, p, λ)}) (⟨f^{p}⟩_{μ} + ∫_{0}^{t} exp (\frac{ps}{c(r, p, λ)}) (μ[(Q_{s}f)^{p/2}])^{2})

for all 0 < ε ≤ f ∈ 𝔻_{∞}(L_{M}), and then for all 0 ≤ f ∈ L^{p}(μ) by the same argument as in Step 1.

The inequality above entails ∥Q_{t}∥_{tail(p,μ)} ≤ exp(−t/c(r, p, λ)). Letting r → r_{0}+ we get (6.20).

Consequently the RTNC is satisfied by (Q_{t}) in L^{p}(μ). By Lemma 3.2(a), there exists ψ ∈ L^{p′}(μ) with ⟨ψ⟩_{μ} = 1 such that λ(R_{λ}^{Q})^{∗}ψ = ψ for some λ > 0. Then Q_{t}^{∗}ψ = ψ for all t ≥ 0. Note that (Q_{t}) is μ-essential irreducible as well as (P_{t}). Thus ψ > 0, μ − a.e. and α := ψμ is the unique invariant measure absolutely continuous/μ.

Step 3. Let us prove \sqrt{ψ} ∈ 𝔻(E) by means of large deviations technique as in [55]. To this end consider

Λ^{Q}(V) := lim sup_{t→∞} \frac{1}{t} log 𝔼^{ℚ_{μ}} exp (∫_{0}^{t} V (X_{s})ds) .

By Hölder’s inequality, for any q ∈ (1, 2λ) fixed and V ∈ bB,

𝔼^{ℚ_{μ}} exp (∫_{0}^{t} V (X_{s})ds) ≤ (𝔼^{ℙ_{μ}} exp [qM_{t} − \frac{q^{2}}{2} ⟨M⟩_{t}])^{1/q}

· (𝔼^{ℙ_{μ}} exp [q′ ∫_{0}^{t} V (X_{s})ds + \frac{q}{2}⟨M⟩_{t}])^{1/q′}

≤ (𝔼^{ℙ_{μ}} exp [q′ ∫_{0}^{t} V (X_{s})ds + \frac{q}{2}⟨M⟩_{t}])^{1/q′}

for exp [qM_{t} − \frac{q^{2}}{2} ⟨M⟩_{t}] being a ℙ_{μ}-local martingale, is a supermartingale. Consequently by Lemma 6.7 and the convexity of Λ(V) on V ∈ bB,

Λ^{Q}(V) ≤ \frac{1}{q′} Λ (V + (q/2)B) ≤ \frac{1}{q′} (\frac{q}{2λ}Λ(λB) + \frac{2λ − q}{2λ} Λ (\frac{2λ}{2λ − q} V )) (6.33)

Thus for any γ ∈ M_{1}(E), the Legendre transformation

(Λ^{Q})^{∗}(γ) := sup_{V ∈bB}(γ(V) − Λ^{Q}(V)) ≥ \frac{2λ − q}{2λq′} Λ^{∗}(γ).

But (Λ^{Q})^{∗}(α) = 0 by [55], (B.23), Proposition B.10 and (B.27). Thus Λ^{∗}(α) < +∞. That implies \sqrt{ψ} ∈ 𝔻(E) and Λ^{∗}(α) = E[\sqrt{ψ}] by [9], Exercise 4.2.63, (4.2.64).

Step 4. The exponential convergence in probability ℚ_{ν} of L_{t} to α (as t goes to infinity) follows from (6.20) and Theorem 3.9.

Step 5. If moreover P_{t} is absolutely continuous/μ, so does (Q_{t}). Thus (6.21) follows from (6.20) by de Pagter-Schep’s Theorem 1.4.

Part (b). By Theorem 2.8, the RUPI of (P_{t}) assumed in (Co6) implies the UPI of (P_{t}). Thus Q_{t} is UPI/μ by Lemma 2.1. Now the exponential convergence (6.22) follows from Hino’s Theorem 1.1.

In the symmetric case, [(6.22)⇒(6.23)] is proved in [16]. □

7 Proofs of the results on RUPI in the discrete time case

7.1 Several lemmas

I learn the following beautiful lemma from Talagrand’s course in Saint-Flour [47](2000) (see Aida [1], [2] for different variants)

Lemma 7.1. (Paley-Zygmund) For a nonnegative random variable X on (Ω, F, ℙ) and for t ∈ (0, 1), q ∈ (1, +∞],

ℙ(X > t𝔼X) ≥ ((1 − t)\frac{𝔼X}{∥X∥_{q}} )^{q′} .  (7.1)

Proof. By Hölder’s inequality,

\begin{aligned}𝔼X & = 𝔼X1_{[X≤t𝔼X]} + 𝔼X1_{[X>t𝔼X]} \\ & ≤ t𝔼X + ∥X∥_{q}ℙ(X > t𝔼X)^{1/q′} \end{aligned}

where follows (7.1).

The following lemma is our main tool which allows us to read all properties in Theorem 2.5 as different faces of the same thing=RUPI .

Lemma 7.2. Let P be a Markov kernel such that μP = μ and p ∈ [1, +∞), q ∈ (1, +∞]. Then for each 0 ≠ f ∈ L^{p}(μ) ⋂ L^{q}(μ) with μ(f) = 0,

χ_{P} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1}^{p} ≤ 4^{p−1}⟨(I − P)f, |f|^{p−1}sgn(f)⟩_{μ}  (7.2)

where sgn(f) = 1_{[f>0]} − 1_{[f<0]} and the convention 0^{0} := 1 is used.

Proof. Fix f ∈ L^{p}(μ) ⋂ L^{q}(μ) with μ(f) = μ(f_{+})−μ(f_{−}) = 0, where f_{±} := (±f)∨0. Summing

\begin{aligned}(f_{+})^{p−1} (f − P f) & = (f_{+})^{p} − (f_{+})^{p−1}P |f| + 2(f_{+})^{p−1}P f_{−} \\ (f_{−})^{p−1} (P f − f) & = (f_{−})^{p} − (f_{−})^{p−1}P |f| + 2(f_{−})^{p−1}P f_{+} \end{aligned}

we obtain

sgn(f)|f|^{p−1}(I − P)f = |f|^{p} − |f|^{p−1}P |f| + 2(f_{+})^{p−1}P f_{−} + 2(f_{−})^{p−1}P f_{+}. (7.3)

By Hölder’s inequality and ∥P ∥_{p,p} = 1,

∫_{E}(|f|^{p} − |f|^{p−1}P |f|)dμ ≥ ∥f∥_{p}^{p} − ∥P |f|∥_{p} · ∥|f|^{p−1}∥_{p′} ≥ 0.

Moreover using μ(f_{+}) = μ(f_{−}) = μ(|f|)/2 we have by Lemma 7.1 (with t = 1/2),

μ (f_{+} > \frac{μ(|f|)}{4}) ≥ (\frac{∥f_{+}∥_{1}}{2∥f_{+}∥_{q}} )^{q′} ≥ (\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′} .

The same holds for μ (f_{−} > μ(|f|)/4). Then

\begin{aligned}∫_{E}(f_{+})^{p−1}P f_{−}dμ & ≥ (\frac{μ(|f|)}{4})^{p} ∫_{E} 1_{[f_{+}>μ(|f|)/4]}P 1_{[f_{−}>μ(|f|)/4]}dμ \\ & ≥ (\frac{μ(|f|)}{4})^{p} χ_{P} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) \end{aligned}

and the same control for ∫_{E}(f_{−})^{p−1}P f_{+}dμ. Substituting them into (7.3) we obtain

4 · χ_{P} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) (\frac{∥f∥_{1}}{4} )^{p} ≤ ⟨sgn(f)|f|^{p−1}, (I − P)f⟩_{μ}

where follows (7.2).

Corollary 7.3. Let P be a Markov kernel such that μP = μ and p ∈ [1, +∞), q ∈ (1, +∞]. Assume that P is RUPI. Then for each 0 ≠ f ∈ L^{q}(μ) with μ(f) = 0,

χ^{(R)} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1} ≤ ∥(I − P)f∥_{1}  (7.4)

∥f∥_{1} ≤ r∥f∥_{q} + \frac{1}{χ^{(R)} ((\frac{r}{4})^{q′})}∥(I − P)f∥_{1}.  (7.5)

where χ^{(R)}(ε) := sup_{λ>0} λ^{2}χ_{R_{λ}}(ε) is the RUPI coefficient of P (w.r.t. μ). If moreover P is symmetric w.r.t. μ, then

⟨|f|⟩_{μ}^{2} ≤ r⟨f^{2}⟩_{μ} + \frac{4}{χ^{(R)} (\frac{r}{16})}⟨f, (I − P)f⟩_{μ}  (7.6)

⟨|f|⟩_{μ}^{2} ≤ r∥f∥_{∞}^{2} + \frac{4}{χ^{(R)} (\frac{\sqrt{r}}{4})}⟨f, (I − P)f⟩_{μ}.  (7.7)

Proof. For each λ > 0, λR_{λ} is Markov with μ as invariant measure, and UPI by Lemma 2.3. Applying Lemma 7.2(7.2) to λR_{λ}, we get for each f ∈ L^{p}(μ) ⋂ L^{q}(μ) non-zero with μ(f) = 0,

\begin{aligned}χ_{λR_{λ}} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1}^{p} & ≤ 4^{p−1}⟨(I − λR_{λ})f, |f|^{p−1}sgn(f)⟩_{μ} \\ & = 4^{p−1}⟨(I − P)R_{λ}f, |f|^{p−1}sgn(f)⟩_{μ}. \end{aligned}  (7.8)

If p = 1, the r.h.s. above is smaller than

∥(I − P)R_{λ}f∥_{1} ≤ ∥R_{λ}∥_{1,1} · ∥(I − P)f∥_{1} ≤ \frac{1}{λ}∥(I − P)f∥_{1}

where (7.4) follows by (7.8). And (7.5) is an immediate consequence of (7.4).

In the symmetric case, the r.h.s. of (7.8) with p = 2 is smaller than (by Cauchy-Schwartz)

4\sqrt{⟨(I − P)f, f⟩_{μ} · ⟨(I − P)R_{λ}f, R_{λ}f⟩_{μ}} ≤ \frac{4}{λ}⟨(I − P)f, f⟩_{μ}

where (7.6) and (7.7) follow from (7.8) respectivement for q = 2 and q = +∞.

7.2 Proof of Proposition 2.6

Inequality (2.16) is contained in Corollary 7.3 (7.4). For establishing the convergence rate (2.17), note that (I − P)Q_{n} = (1/n)(I − P^{n}), where Q_{n} := \frac{1}{n} ∑_{k=0}^{n−1} P^{k}. For any f with ∥f∥_{∞} ≤ 1, since ∥Q_{n}(f) − μ(f)∥_{∞} ≤ 2, we have by (2.16) or (7.4) (with q = +∞)

χ^{(R)} (\frac{∥Q_{n}(f) − μ(f)∥_{1}}{4 × 2} ) ∥Q_{n}f − μ(f)∥_{1} ≤ ∥(I − P)(Q_{n}f − μ(f))∥_{1} ≤ \frac{2}{n}.

Whence (2.17) follows. □

7.3 Proof of Theorem 2.5

Recall that ∥ · ∥_{p} denotes the norm in L^{p}(E, μ). For clarity, let us denote the constant c(r) in the weak spectral gap inequality in the property (Pk) by c_{k}(r) for k = 4, 5, 6, 7, 8, 9.

We shall divide the proof of Theorem 2.5 into four parts:

Part 1.

(P 1) ⟹ (P 2) ⟹ (P 3) ⟹ (P 4).

Part 2.

[(P 4) ⇔ (P 4b)] ⟹ [(P 5) ⇔ (P 5b)] ⟹ [(P 6) ⇔ (P 6b)] ⟹ (P 9b).

Part 3.

(P 1) ⟹ [(P 7) ⇔ (P 7b)] ⟹ [(P 8) ⇔ (P 8b)] ⟹ [(P 9) ⇔ (P 9b)].

Part 4.

(P 9) ⟹ [(P 10) ⇔ (P 10b)] ⟹ (P 11b) ⟹ (P 11) ⟹ (P 1).

Proof. of Part 1.

(P 1) ⟹ (P 2): trivial, as 2P^{σ} ≥ P.

(P 2) ⟹ (P 3): This is the beginning point of this theorem. By (7.8) with p = 2 in the proof of Corollary 7.3, we have for any q ∈ (1, +∞], λ > 0 and f ∈ L_{0}^{p}(μ),

χ_{λR_{λ}^{σ}} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1}^{2} ≤ 4⟨(I − P^{σ})R_{λ}^{σ}f, f⟩_{μ} ≤ \frac{4}{λ}⟨(I − P^{σ})f, f⟩_{μ}

where R_{λ}^{σ} = (λ+1−P^{σ})^{−1} and the last inequality follows by spectral decomposition. Whence inequality (2.3) is true with

Θ_{q}(r) = sup_{λ>0} λ^{2}χ_{R_{λ}^{σ}} ((\frac{r}{4})^{q′}) .  (7.9)

(P 3) ⟹ (P 4): (P4) is a particular case of (P3) with q = 2.

Proof. of Part 2. The proof in this part is easy.

(P4)⇔(P4b): For (P4) ⟹ (P4b), notice that for every sequence (f_{n}) satisfying ∥f_{n}∥_{2} ≤ 1, μ(f_{n}) = 0 and ⟨f_{n}, (I − P)f_{n}⟩_{μ} → 0. We have by the assumed (2.3),

lim sup_{n→∞} ∥f_{n}∥_{1}^{2} ≤ r.

As r can be chosen arbitrary by the assumed (2.3), ∥f_{n}∥_{1} → 0, the desired (P4b).

We now show (P4b) ⟹ (P4) by absurd. Assume in contrary that (P4) were false. Then (∃r_{0} > 0)(∀n ≥ 1)(∃f_{n} ∈ L_{0}^{2}(μ), ∥f_{n}∥_{2} = 1):

∥f_{n}∥_{1}^{2} ≥ r_{0} + n⟨f_{n}, (I − P)f_{n}⟩_{μ}.

Since ∥f_{n}∥_{1} ≤ ∥f_{n}∥_{2} = 1, this yields lim_{n}⟨f_{n}, (I − P)f_{n}⟩_{μ} = 0. By the assumed (P4b), ∥f_{n}∥_{1} → 0, a contradiction with the inequality just obtained above.

By the same proof, we have (Pk) ⇔ (Pkb) for k = 5, 6, 7, 8, 9.

(P4) ⟹ (P5): trivial, moreover c_{5}(r) ≤ c_{4}(r).

(P5b) ⟹ (P6b): easy, for (∥f_{n}∥_{1} → 0) + (∥f_{n}∥_{∞} ≤ 1) implies ∥f_{n}∥_{p} → 0.

(P6b) ⟹ (P9b): For every sequence (f_{n}) such that ∥f_{n}∥_{∞} ≤ 1, μ(f_{n}) = 0 and ∥(I − P)f_{n}∥_{∞} → 0, we have

E^{σ}[f_{n}] = ⟨f_{n}, (I − P)f_{n}⟩_{μ} ⟶ 0.

Thus f_{n} ⟶ 0 in L^{1}(μ) by (P6b), then in L^{p}(ν) by dominated convergence.

Proof. of Part 3. This part is again very easy.

(P1) ⟹ (P7): This is contained in Corollary 7.3.

(P7) ⟹ (P8): Trivial with c_{8}(r) ≤ c_{7}(r).

(P8) ⟹ (P9): trivial with c_{9}(r) ≤ c_{8}(r).

Proof. of Part 4.

(P9) ⟹ (P10): Let Q_{n} := (1/n) ∑_{k=0}^{n−1} P^{k}. We have (I − P)Q_{n} = (I − P^{n})/n. For any f ∈ L^{∞} with ∥f∥_{∞} ≤ 1, applying the assumed inequality (2.8) in (P9) to Q_{n}(f − μ(f)), we obtain

∥Q_{n}f − μ(f)∥_{1} ≤ r∥f − μ(f)∥_{∞} + c_{9}(r)\frac{1}{n}∥(I − P^{n})[f − μ(f)]∥_{∞} ≤ 2r + \frac{2c_{9}(r)}{n}.

Letting n → ∞, we get (P10) (as r > 0 is arbitrary).

(P10) ⟹ (P10b): Define θ ∈ (0, 1) and a ∈ (p, +∞) by

θ = \frac{1}{p} − \frac{1}{q}, \frac{1}{p} = \frac{θ}{1} + \frac{1 − θ}{a}.

Then

\frac{1}{q} = \frac{θ}{∞} + \frac{1 − θ}{a}.

Since ∥Q_{n} − μ(·)∥_{a,a} ≤ 2 (for all a ∈ [1, +∞]) in the invariant measure case, we have by Riesz-Thorin’s interpolation theorem

∥Q_{n} − μ(·)∥_{q,p} ≤ ∥Q_{n} − μ(·)∥_{∞,1}^{θ} · ∥Q_{n} − μ(·)∥_{a,a}^{1−θ} ⟶ 0.

Whence (P10b) follows.

(P10b) ⟹ (P10): trivial.

(P10b) ⟹ (P11b): By the same argument as in the classical implication that the Césaro summability is stronger than the Abel summability.

(P11b) ⟹ (P11): trivial.

It remains to close the cycle of equivalence in Theorem 2.5 by

(P11) ⟹ (P1): Since for any A, B ∈ B,

|⟨1_{A}, λR_{λ}1_{B}⟩_{μ} − μ(A) · μ(B)| ≤ ∥λR_{λ} − μ(·)∥_{∞,1},

by the definition of the UPI coefficient χ_{λR_{λ}}(ε), we have for any ε ∈ (0, 1),

χ_{λR_{λ}}(ε) ≥ ε^{2} − ∥λR_{λ} − μ(·)∥_{∞,1}

which is positive for sufficiently small λ by (P11). But by Lemma 2.2 and and the equivalence of {λR_{λ}(x, ·); λ > 0} (x being fixed), we have χ_{λR_{λ}}(ε) > 0 for all λ > 0.

7.4 A lemma for Theorem 2.7

Lemma 7.4. Let P be a Markov kernel such that μP = μ and q ∈ (1, +∞]. Assume that P is RUPI (/μ). Then for any sequence (f_{n})_{n∈ℕ} ⊂ L^{1}(μ) satisfying

• ∥(I − P)f_{n}∥_{1} → 0 and ⟨f_{n}⟩_{μ} = 0;

• {f_{n}}_{n∈ℕ} is uniformly integrable; then f_{n} → 0 in probability μ (then in L^{1}(μ)).

Proof. This property is a refinement of (P7) in Theorem 2.5. For proving it we shall require to refine the inequalities in Lemma 7.1 and Lemma 7.2. Its proof is divided into four steps.

Step 1. As (f_{n})_{n≥1} is uniformly integrable, by Vallée Poussin’s theorem, there is some C^{1}-convex, even and nonnegative function Φ : ℝ → ℝ^{+} satisfying

ϕ(t) := Φ′(t) is strictly increasing on ℝ^{+}, Φ(0) = ϕ(0) = 0, lim_{t→+∞} ϕ(t) = +∞,

such that

sup_{n≥1} ∫_{E} Φ(f_{n})dμ ≤ \frac{1}{2}

(this version is slightly stronger than the usual one such as [39], p.3, Theorem 2, but it can be obtained in the same way).

An important fact is: the Fenchel-Legendre transformation

Ψ(s) := sup_{t≥0}(|s|t − Φ(t)), ∀s ∈ ℝ

is an even convex function given by

Ψ(s) = ∫_{0}^{s} ψ(u)du, ψ(s) := ϕ^{−1}(s) (inverse function).

Step 2. In this step we extend Lemma 7.1.

Claim 1. Let Φ be an even convex function as in Step 1. If g ≥ 0 and μ(g) > 0, then

μ(g > t⟨g⟩_{μ}) ≥ \frac{1}{Ψ (\frac{∥g∥_{Φ}}{(1−t)∥g∥_{1}} )}, ∀t ∈ (0, 1)  (7.10)

where

∥g∥_{Φ} := sup {∫_{E} ghdμ| ∫_{E} Ψ(|h|)dμ ≤ 1}

is the Orlicz norm of g in the Orlicz space L_{Φ}(μ).

Recall the following inequality (see [39])

N_{Φ}(g) ≤ ∥g∥_{Φ} ≤ 2N_{Φ}(g)

where N_{Φ}(g) := inf{c > 0; ⟨Φ(g/c)⟩_{μ} ≤ 1} is the gauge norm.

For showing (7.10), using

μ(g) = ∫_{[g≤tμ(g)]} gdμ + ∫ g1_{[g>t⟨g⟩_{μ}]}dμ < tμ(g) + ∥g∥_{Φ} · N_{Ψ}(1_{[g>t⟨g⟩_{μ}]})

(where the last inequality follows by a generalized Hölder inequality in [39], p.62), we get

N_{Ψ}(1_{[g>t⟨g⟩_{μ}]}) > \frac{(1 − t)μ(g)}{∥g∥_{Φ}}.

Then by definition of the gauge norm N_{Ψ} (and Ψ(0) = 0),

1 < ∫_{E} Ψ (\frac{1_{[g>t⟨g⟩_{μ}]}∥g∥_{Φ}}{(1 − t)μ(g)} ) dμ = Ψ (\frac{∥g∥_{Φ}}{(1 − t)μ(g)}) μ[g > t⟨g⟩_{μ}]

where follows (7.10).

Step 3. In this step we extend Lemma 7.2 as follows.

Claim 2. Let Φ and Ψ as in Step 1. Then for any p ∈ [1, +∞) and f ∈ L_{0}^{p}(μ) ⋂ L_{Φ}(μ),

χ_{P} (\frac{1}{Ψ (\frac{4∥f∥_{Φ}}{∥f∥_{1}} )}) · ∥f∥_{1}^{p} ≤ 4^{p−1}⟨(I − P)f, |f|^{p−1}sgn(f)⟩_{μ}.  (7.11)

Its proof, based on (7.10), is completely parallel to that of Lemma 7.2, so omitted.

Step 4. Applying (7.11) with p = 1 to P given by λR_{λ}, and to f given by f_{n} in Step 1, we obtain (since ∥f_{n}∥_{Φ} ≤ 2N_{Φ}(f_{n}) ≤ 1),

χ_{λR_{λ}} (\frac{1}{Ψ (\frac{4}{∥f_{n}∥_{1}} )}) · ∥f_{n}∥_{1} ≤ ∥(I − λR_{λ})f_{n}∥_{1} ≤ \frac{1}{λ}∥(I − P)f_{n}∥_{1}.

Whence ∥f_{n}∥_{1} ⟶ 0, the desired result.

Remarks: One naturally suspect if the uniform integrability of (f_{n}) in this lemma could be substituted by the boundedness in L^{1}(μ), to which we don’t know the answer. But we know a much simpler proof of this lemma in the symmetric case.

Indeed in the symmetric case that P = P^{∗}, we have

⟨(I − P)f, g⟩_{μ} = ∫∫ μ(dx)P (x, dy) (f(y) − f(x)) (g(y) − g(x)) .

Hence

⟨(I − P) arctan f, arctan f⟩_{μ} ≤ ⟨(I − P)f, arctan f⟩_{μ}.  (7.12)

Hence for the sequence (f_{n}) verifying the conditions of Lemma 7.4, g_{n} := arctan f_{n}− ⟨arctan f_{n}⟩_{μ} satisfies

∥g_{n}∥_{∞} ≤ π, ⟨(I − P)g_{n}, g_{n}⟩_{μ} → 0.

By (P6) in Theorem 2.5, g_{n} → 0 in probability. Finally let c ∈ [−π/2, π/2] be a limit point of (⟨arctan f_{n}⟩_{μ})_{n≥1} or limit of some subsequence (⟨arctan f_{n_{k}}⟩_{μ})_{k≥1}. Then f_{n_{k}} converges in probability to tan c, then in L^{1}(μ) by the uniform integrability. Hence tan c = lim_{k→∞}⟨f_{n_{k}}⟩_{μ} = 0, i.e., c = 0. In other words arctan f_{n} ⟶ 0 in probability, the desired claim.

But this argument depends heavily on the contraction property (7.12) which might be false in the non-symmetric case.

7.5 Proof of Theorem 2.7

Now though ν is different from μ, we note however

• the WSGP (P9b) w.r.t. ν and μ are equivalent;

• (P7) ⟹ (P8) becomes now very difficult, but it is obvious that (P7) ⟹ (P9b);

• the uniform ergodic theorems (P10) and (P11) w.r.t. ν and μ are equivalent.

With those points in mind, we see from the proof of Theorem 2.5 that for proving Theorem 2.7, we have only to establish the implication ”(P1) ⟹ (P8b)”. For that implication let (f_{n}) be a sequence in L^{p}(ν) satisfying the condition of (P8b): ⟨f_{n}⟩_{μ} = 0, ∥f_{n}∥_{L^{p}(ν)} ≤ 1 and (I − P)f_{n} ⟶ 0 in L^{p}(ν).

By Hölder’s inequality,

∫_{E} |(I − P)f_{n}|dμ = ∫_{E} |(I − P)f_{n}|ψdν ≤ ∥(I − P)f_{n}∥_{L^{p}(ν)} · ∥ψ∥_{L^{p′}(ν)} ⟶ 0.

And by Hölder’s inequality again, {f_{n}} is bounded in L^{1}(μ) and

lim sup_{μ(A)→0} ∫_{A} |f_{n}|dμ ≤ lim sup_{μ(A)→0} ∥f_{n}∥_{L^{p}(ν)} · ∥1_{A}ψ∥_{L^{p′}(ν)} = 0.

Hence {f_{n}} is μ-uniformly integrable. Therefore by Lemma 7.4, f_{n} → 0 in measure μ, then in measure ν. Thus f_{n} → 0 in L^{1}(ν) (by the boundedness in L^{p}(ν)), the desired claim in (P8b).

8 Proofs of the results on RUPI in the continuous time case

This section is devoted to the proofs of the main results on RUPI in the continuous time case: Theorem 2.8 and 2.10

8.1 Proof of Theorem 2.8

As in the proof of Theorem 2.5, for clarity, let us denote the constant c(r) in the weak spectral gap inequality in the property (Pk) by c_{k}(r) for k = 4, 5, 6, 7, 8, 9.

(C1)⇔(C2): trivial for χ_{π}(ε/μ) = χ_{π^{∗}}(ε/μ).

(C1) ⟹ (C3): As λR_{λ} is UPI, we have by the key Lemma 7.2 that for any 0 ≠ f ∈ L_{0}^{p}(μ) ⋂ L_{0}^{q}(μ),

χ_{λR_{λ}} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1}^{p} ≤ 4^{p−1}⟨(I − λR_{λ})f, |f|^{p−1}sgn(f)⟩_{μ}.  (8.1)

Letting p = 1 and noting (I − λR_{λ})f = −LR_{λ}f, we have for all 0 ≠ f ∈ L_{0}^{q}(μ) ⋂ 𝔻_{L^{1}(μ)}(L),

χ_{λR_{λ}} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1} ≤ ⟨−LR_{λ}f, sgn(f)⟩_{μ} ≤ ∥R_{λ}∥_{L^{1}(μ)→L^{1}(μ)} · ∥Lf∥_{L^{1}(μ)}.

As ∥R_{λ}∥_{L^{1}(μ)→L^{1}(μ)} ≤ 1/λ, (2.19) holds with

Θ_{q}(r) = sup_{λ>0} χ_{λ^{2}R_{λ}} ((\frac{r}{4})^{q′}) = χ^{(R)} ((\frac{r}{4})^{q′}) .  (8.2)

(C3) ⟹ (C6b): For (f_{n}) satisfying ∥f_{n}∥_{∞} ≤ 1 and ∥Lf_{n}∥_{∞} ⟶ 0, we have by (2.19) in (C3) (with q = +∞),

Θ_{∞}(⟨|f_{n}|⟩_{μ})⟨|f_{n}|⟩_{μ} ≤ ∥Lf_{n}∥_{L^{1}(μ)} ≤ ∥Lf_{n}∥_{∞} ⟶ 0.

Hence ⟨|f_{n}|⟩_{μ} → 0, then in measure ν for μ ∼ ν. But ∥f_{n}∥_{∞} ≤ 1, so f_{n} → 0 in L^{1}(ν).

(C1) ⟹ (C4): Since the Markov operator λR_{λ} is bounded and UPI on L^{p}(ν) for some λ > 0, and has μ = ψν as invariant measure, applying Theorem 2.7(P7) to λR_{λ}, we have by noting that −LR_{λ} = I − λR_{λ},

∥f∥_{L^{1}(ν)} ≤ r∥f∥_{L^{p}(ν)}+c(r)∥(I−λR_{λ})f∥_{L^{p}(ν)} ≤ r∥f∥_{L^{p}(ν)}+c(r)∥Lf∥_{L^{p}(ν)}·∥R_{λ}∥_{L^{p}(ν)→L^{p}(ν)}

where (C4) follows.

(Ck)⇔(Ckb) for k = 4, 5, 6: the same proof as that of (P4)⇔(P4b) in Section 4.

(C4b) ⟹ (C5b): obvious.

(C5) ⟹ (C6): trivial and c_{6}(r) ≤ c_{5}(r).

(C6) ⟹ (C7): For any f ∈ L^{∞}(ν) with ∥f∥_{∞} ≤ 1, Q_{T} f := (1/T) ∫_{0}^{T} P_{t}fdt ∈ 𝔻_{q}(L) and

LQ_{T} f = \frac{1}{T} (P_{T} f − f) .

Hence applying (C6) to Q_{T} f − μ(f), we get

∥Q_{T} f − μ(f)∥_{L^{1}(ν)} ≤ r∥Q_{T} f − μ(f)∥_{∞} + c_{6}(r)\frac{1}{T} ∥(P_{T} f − f)∥_{∞} ≤ 2r + \frac{2c_{6}(r)}{T}

where (C7) follows.

(C7)⇔(C7b): We note that the uniform ergodic theorem (C7) w.r.t. ν is equivalent to that w.r.t. μ. Hence (C7b)Rightarrow(C7). And (C7) w.r.t. μ implies (C7b) by the interpolation as in the proof of (P10) ⟹ (P10b) in Theorem 2.5.

(C6) ⟹ (C8): Applying (C6) to f = λR_{λ}g − μ(g) where ∥g∥_{∞} ≤ 1, we obtain

\begin{aligned}∥λR_{λ}g − μ(g)∥_{L^{1}(ν)} & ≤ r∥λR_{λ}g − μ(g)∥_{∞} + c(r)∥L(λR_{λ}g − μ(g))∥_{∞} \\ & ≤ 2r + c_{6}(r)λ∥(I − λR_{λ})g∥_{∞} ≤ 2r + 2c_{6}(r)λ \end{aligned}

where (C8) follows.

(C8) ⟹ (C8b): Note that (C8) w.r.t. ν or w.r.t. μ are equivalent. And (C8) w.r.t. μ implies (C8b) by interpolation as in the proof of (P10) ⟹ (P10b) in Theorem 2.5.

vskip10pt (C8) ⟹ (C8b): for (C8b) implies (C7) w.r.t. μ which is equivalent to (C7) w.r.t. ν.

Now to close the cycle of equivalence in Theorem 2.8, it remains to show

(C7) ⟹ (C1): For each ε ∈ (0, 1) and all A, B ∈ B with μ(A) ∧ μ(B) ≥ ε, we have

⟨1_{A}, Q_{T} 1_{B}⟩_{ν} ≥ μ(A)μ(B) − ∥Q_{T} − μ(·)∥_{L^{∞}→L^{1}(ν)} ≥ ε^{2} − ∥Q_{T} − μ(·)∥_{L^{∞}→L^{1}(ν)}.

Hence χ_{Q_{T}} (ε/ν) > 0 for T > 0 large enough (by (C7)). Hence

χ_{R_{λ}}(ε/ν) ≥ χ_{Q_{T}} (ε/ν) · T e^{−λT} > 0.

(C8) ⟹ (C1): same as in the proof of (P11) ⟹ (P1) in Theorem 2.5. □

8.2 Proof of Proposition 2.9

Note that for any f ∈ L^{∞} with ∥f∥_{∞} ≤ 1,

∫_{0}^{t} P_{s}fds ∈ 𝔻_{2}(L), L ∫_{0}^{t} P_{s}fds = P_{t}f − f.

In the implication of (C1) ⟹ (C3) above, we have indeed derived (see (8.2))

χ^{(R)} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1} ≤ ∥Lf∥_{1}, ∀0 ≠ f ∈ L_{0}^{q}(μ) ⋂ 𝔻_{1}(L)  (8.3)

which is the desired (2.27). Applying it with q = +∞ to (1/t) ∫_{0}^{t} P_{s}fds − μ(f), we obtain the convergence rate (2.28) as in Proposition 2.6. □

8.3 Proof of Theorem 2.10

Lemma 8.1. Assume that (P_{t})_{t∈ℝ^{+}} be a C_{0}-semigroup of contractions on L^{2}(μ), whose generator L satisfies the sector condition (2.29). Then

(a) there is some constant M = M(κ) > 0 (depending only on κ) such that for any t > 0, ∥LP_{t}∥_{L^{2}(μ)→L^{2}(μ)} ≤ M/t (and M = 1 in the symmetric case);

(b) for any λ > 0 and for all f ∈ 𝔻_{2}(L),

⟨(I − λR_{λ})f, f⟩_{μ} ≤ \frac{κ^{2}}{λ} ⟨−Lf, f⟩_{μ}.  (8.4)

Proof. (a) This part is well known, see Kato [25], Chap. IX, Section 1.

(b) We assume at first that ⟨−Lf, f⟩_{μ} ≥ ε⟨f, f⟩_{μ} for some ε > 0 (for all f ∈ 𝔻_{2}(L)). By Lumer-Phillips theorem, ∥P_{t}∥_{2} ≤ e^{−εt} for all t ≥ 0.

Let R_{λ} := (λ − L)^{−1} for λ ≥ 0 (R_{0} is the Poisson operator which is bounded under the extra condition assumed above). Applying sector condition (2.29) to f = R_{0}h_{1}, g = R_{0}h_{2}, we see that R_{0} satisfies the sector condition (2.29) with the same constant κ, as well as L.

Now for any f ∈ 𝔻_{2}(L), we have by (2.29),

⟨(I − λR_{λ})f, f⟩_{μ} = ⟨−LR_{λ}f, f⟩_{μ} ≤ κ\sqrt{⟨−Lf, f⟩_{μ} · ⟨−LR_{λ}f, R_{λ}f⟩_{μ}}  (8.5)

Let g := −Lf. Since (R_{0} − R_{λ})g = λR_{0}R_{λ}g, we have

⟨R_{λ}g, R_{0}R_{λ}g⟩_{μ} = \frac{1}{λ}⟨R_{λ}g, (R_{0} − R_{λ})g⟩_{μ}

≤ \frac{1}{λ}⟨R_{λ}g, R_{0}g⟩_{μ}

≤ \frac{κ}{λ}\sqrt{⟨R_{0}R_{λ}g, R_{λ}g⟩_{μ} · ⟨R_{0}g, g⟩_{μ}}

where the last inequality follows by the sector condition possessed by R_{0}. Therefore

⟨−LR_{λ}f, R_{λ}f⟩_{μ} = ⟨R_{λ}g, R_{λ}R_{0}g⟩_{μ} ≤ \frac{κ^{2}}{λ^{2}} ⟨R_{0}g, g⟩_{μ} = \frac{κ^{2}}{λ^{2}} ⟨f, −Lf⟩_{μ}.

Substituting it into (8.5) we obtain (8.4) under the extra assumption.

In the general case, for each ε > 0, consider L_{ε} := L − ε, which satisfies again (2.29) with the same constant κ. Noting that (λ − L_{ε})^{−1} = R_{λ+ε}, we get by the first case,

⟨(I − λR_{λ+ε})f, f⟩_{μ} ≤ \frac{κ^{2}}{λ} ⟨(−L + ε)f, f⟩_{μ}, ∀f ∈ 𝔻_{2}(L).

Now letting ε → 0+, we obtain (8.4).

Proof of Theorem 2.10. RUPI ⟹ (E1): By Lemma 7.2 with p = 2 and by Lemma 8.1, we have

χ_{λR_{λ}} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1}^{2} ≤ 4⟨(I − λR_{λ})f, f⟩_{μ} ≤ \frac{4κ^{2}}{λ} ⟨−Lf, f⟩_{μ}

for all f ∈ 𝔻_{2}(L) with ⟨f⟩_{μ} = 0. Hence

χ^{(R)} ((\frac{∥f∥_{1}}{4∥f∥_{q}} )^{q′}) · ∥f∥_{1}^{2} ≤ 4κ^{2}⟨−Lf, f⟩_{μ}  (8.6)

i.e., inequality (2.30) in (E1) holds with

Θ_{q}(r) = \frac{1}{4κ^{2}} χ^{(R)} ((\frac{r}{4})^{q′}) .  (8.7)

(E1) ⟹ (E2): This is obvious with c_{2}(r) = 1/Θ_{q}(\sqrt{r}).

(E2) ⟹ (E3): If p = 1, (E3) is a particular case of (E2). Now for p ∈ (1, +∞),

∥f∥_{p} ≤ ∥f∥_{1}^{1/p} · ∥f∥_{∞}^{1/p′} ≤ \frac{r}{2p′} ∥f∥_{∞} + \frac{1}{p} (\frac{r}{2})^{−p/p′} ∥f∥_{1}.

By (E2) with r given by (\frac{1}{p} (\frac{r}{2})^{−p/p′})^{−1} /2, we get inequality (2.32) in (E3).

(E3) ⟹ (E4): For any r, t > 0 and f with ∥f∥_{∞} ≤ 1, we have by (E3) and by Lemma 8.1

∥P_{t}f − μ(f)∥_{1}^{2} ≤ 4r + c_{3}(r)⟨−LP_{t}f, P_{t}f⟩_{μ} ≤ 4r + c(r)∥LP_{t}f∥_{2} ≤ 4r + \frac{Mc(r)}{t}

where it follows lim sup_{t→∞} ∥P_{t} − μ(·)∥_{∞,1} ≤ 4r. As r > 0 is arbitrary, we get (E4).

(E4)⇔(E5): The direct implication ” ⟹ ” follows by interpolation as in (P10) ⟹ (P10b) in the proof of Theorem 2.5. The inverse is trivial.

(E4)⇒(E6): it follows by (2.14).

(E4) ⟹ (E7): For any ε > 0, we have

χ_{P_{t}}(ε) ≥ ε^{2} − ∥P_{t} − μ(·)∥_{∞,1}

where (E7) follows by (E4).

(E7) ⟹ (E1): For each ε > 0 and A, B ∈ B with μ(A) ∧ μ(B) ≥ ε, and noting that μ(P_{s}^{∗}1_{B}) = μ(B), we have by Lemma 7.1 (with q = +∞) that

μ(P_{s}^{∗}1_{B} > ε/2) ≥ μ(P_{s}^{∗}1_{B} > μ(P_{s}^{∗}1_{B})/2) ≥ \frac{1}{2}μ(B) ≥ \frac{ε}{2}.

Hence

⟨P_{t+s}1_{A}, 1_{B}⟩_{μ} = ⟨P_{t}1_{A}, P_{s}^{∗}1_{B}⟩_{μ} ≥ \frac{ε}{2}χ_{P_{t}}(ε/2).

We have so proved

χ_{P_{t+s}}(ε) ≥ \frac{ε}{2}χ_{P_{t}} (\frac{ε}{2}) , ∀t, s ≥ 0.

Hence (E7) ⟹ (E1). □

8.4 Proof of Proposition 2.11

By applying (2.27) with q = +∞, to P_{t}f − μ(f) where ∥f∥_{∞} ≤ 1, we get

χ^{(R)} (\frac{∥P_{t}f − μ(f)∥_{1}}{8} ) · ∥P_{t}f − μ(f)∥_{1} ≤ ∥ − LP_{t}f∥_{2} ≤ \frac{M(κ)}{t}

where M(κ) is the universal constant in Lemma 8.1(a). Thus (2.33) holds with M = M(κ).

9 Probabilistic Interpretation of the RTNC: proof of Theorems 3.9 and 3.8

9.1 Preparations

As in Deuschel and Stroock [9], given a non-empty family A of initial measures γ ≪ μ, we introduce the A-uniform Cramèr functional

Λ_{A}(V) := lim sup_{t→∞} \frac{1}{t} log sup_{γ∈A} 𝔼^{γ} exp (tL_{t}(V))
= lim sup_{t→∞} \frac{1}{t} log sup_{γ∈A} γ(P_{t}^{V} 1)

(9.1)

for any V ∈ bB, where

P_{t}^{V} f(x) := 𝔼^{x}f(X_{t}) exp (tL_{t}(V))  (9.2)

is the Feynman-Kac semigroup associated with potential V. Let (bB)^{#} be its algebrical dual (i.e., the space of all real linear forms on bB), equipped with the weak topology σ ((bB)^{#}, bB). A subset of (bB)^{#} is said to be measurable if it belongs to σ(β ↦ β(V); V ∈ bB).

The space M_{b}(E) of bounded (signed) σ-addive measures on (E, B) will be identified as a subspace of (bB)^{#}, which is dense for σ ((bB)^{#}, bB).

Consider the Fenchel-Legendre transformation of Λ_{A},

(Λ_{A})^{∗}(β) := sup {β(V) − Λ_{A}(V)| V ∈ bB} , ∀β ∈ (bB)^{#}.  (9.3)

It is obvious that (Λ_{A})^{∗} takes values in [0, +∞] (for Λ_{A}(0) = 0) and it is convex on (bB)^{#}.

Since (bB)^{#}, σ ((bB)^{#}, bB)) is the projective limit of a family of finite dimensional spaces (see Dawson and Gärtner [8], we have

Lemma 9.1. (due to [8]) For any closed and measurable F in (bB)^{#}, σ ((bB)^{#}, bB)),

lim sup_{t→∞} \frac{1}{t} log sup_{γ∈A} ℙ_{γ} (L_{t} ∈ F) ≤ − inf_{β∈F}(Λ_{A})^{∗}(β).  (9.4)

Moreover the rate function (Λ_{A})^{∗} is inf-compact on (bB)^{#}, σ ((bB)^{#}, bB)), i.e., ∀L ≥ 0, the level set [(Λ_{A})^{∗} ≤ L] is compact.

Note: In [8] only the case where A is a singleton is studied. But the extension to the uniform case is completely parallel.

The question for the exponential convergence is reduced to prove the existence and the uniqueness of invariant measure α ≪ μ and to establish

inf_{γ∉N^{#}(α)}(Λ_{A})^{∗}(β) > 0  (9.5)

for any open measurable neighborhood N^{#}(α) of α in (bB)^{#}. Since the infimum at the l.h.s of (9.5) must be attained by the inf-compactness of (Λ_{A})^{∗}, hence for (9.5), we have only to show for β ∈ (bB)^{#},

(Λ_{A})^{∗}(β) = 0 iff β = α.  (9.6)

The aim of the following lemma is to isolate a sufficient condition for (9.6).

Lemma 9.2. Assume (A1) and

(SBC) (Small and Bounded Continuity ) there exists some δ > 0 such that for any sequence (V_{n})_{n∈ℕ} ⊂ bB decreasing pointwise to zero and satisfying ∥V_{0}∥ := sup_{x∈E} |V_{0}(x)| ≤ δ, it holds that Λ_{A}(V_{n}) → 0.

Then there exists a unique invariant probability measure α ≪ μ such that (9.6) holds. In particular the uniform on A exponential convergence in probability (9.5) is true.

Proof. Step 1. By [55], (B.27), Λ_{μ}^{0}(V) ≤ Λ_{A}(V) ≤ Λ_{μ}^{∞}(V) for any V ∈ bB (with the notations therein). Thus by Proposition B.9 and B.10 (respectively for 𝕋 = ℕ or ℝ^{+}), for β ∈ M_{b}(E),

(Λ_{A})^{∗}(β) = \{ \begin{matrix}J_{μ}(β); if β ∈ M_{1}(E); \\ + ∞; if β ∈ M_{b}(E)\M_{1}(E) \end{matrix}  (9.7)

where J_{μ}(β) is the modified Donsker-Varadhan entropy defined in [55] (see (5.2a,b) and Proposition b.9 and B.10 there for explicit expression). Moreover by (B.23) in [55], for β ∈ M_{1}(E), J_{μ}(β) = 0 iff β ≪ μ and βP_{t} = β. Such absolute continuous invariant probability measure, if exists, is unique by the essential irreducibility assumption (A1) (by an elementary argument, see the point (5) in the proof of Theorem 5.1 in [55]).

Step 2. We now show that for β ∈ (bB)^{#}, if (Λ_{A})^{∗}(β) < +∞, then

1) β(V) ≥ 0 for any V ≥ 0; and

2) β(1_{E}) = 1.

Indeed for the first claim, if in contrary β(V_{0}) < 0 for some V_{0} ≥ 0, then using the fact that Λ_{A}(aV_{0}) ≤ 0 for any a ≤ 0, we would get

(Λ_{A})^{∗}(β) ≥ sup_{a≤0}[aβ(V_{0}) − Λ_{A}(aV)] ≥ sup_{a≤0} aβ(V_{0}) = +∞.

This contradiction proves the first claim.

Note that Λ_{A}(c) = c for any constant c ∈ ℝ. Thus

+∞ > (Λ_{A})^{∗}(β) ≥ sup_{c∈ℝ}[β(c) − Λ_{A}(c)] = sup_{c∈ℝ}c(β(1) − 1)

where follows β(1) = 1.

Step 3. By applying (9.4) in Lemma 9.1 to the closed set F = (bB)^{#}, we see that its l.h.s. is zero. Hence

inf_{β∈(bB)^{#}}(Λ_{A})^{∗}(β) = 0.

This infimum should be attained by the inf-compactness of (Λ_{A})^{∗}.

Now let β ∈ (bB)^{#} satisfy (Λ_{A})^{∗}(β) = 0. β must be a positive linear form and β(1) = 1 by Step 2. For any sequence (V_{n})_{n≥0} in bB decreasing pointwise to zero with ∥V_{0}∥ ≤ 1, we have

lim_{n→∞} β(V_{n}) ≤ lim_{n→∞} \frac{1}{δ} [(Λ_{A})^{∗}(β) + Λ_{A}(δV_{n})] = lim_{n→∞} \frac{1}{δ}Λ_{A}(δV_{n}) = 0

where the last equality is our assumption (SBC). Thus by Daniel’s integral representation theorem, β is a σ-additive measure, then a probability measure.

In that case we have (Λ_{A})^{∗}(β) = J_{μ}(β) = 0. Thus β is an absolute continuous invariant measure of (P_{t}) and it is the unique such measure by Step 1.

For verifying the crucial condition (SBC) of Λ_{A}, we apply the following lemma which is an extension of [55], Proposition 3.8.

Lemma 9.3. Let r_{sp}(π) denote the spectral radius of positive operator π on L^{p}(μ). Given a sequence of positive bounded operators (π_{n})_{n∈ℕ̄:=ℕ∪{∞}} on L^{p}(μ), assume that

(i) π_{n}f → π_{∞}f in L^{p} for any f ∈ L^{∞}; or instead

(i^{∗}) π_{n}^{∗}f → π_{∞}^{∗}f in L^{p′} for any f ∈ L^{∞};

(ii) there exists some positive operator π such that π_{n} ≤ π;

(iii) ∥π∥_{tail} < inf_{n∈ℕ} r_{sp}(π_{n}).

Then

lim sup_{n→∞} r_{sp}(π_{n}) ≤ r_{sp}(π_{∞}).  (9.8)

If moreover π_{∞} is μ-essentially irreducible, then lim_{n→∞} r_{sp}(π_{n}) = r_{sp}(π_{∞}).

Proof. Note that, if (9.8) is true under [(i)′ + (ii) + (iii)], then applying it to (π_{n}^{∗}) and reminding the fact that r_{sp}(π) = r_{sp}(π^{∗}), we get (9.8) under [(i) + (ii) + (iii)]. Hence, we only need to prove (9.8) under [(i)′ + (ii) + (iii)].

By conditions (ii) and (iii), ∥π_{n}∥_{tail} ≤ ∥π∥_{tail} < r_{sp}(π_{n}). Then by Lemma 3.2 there exist ϕ_{n} ≥ 0 μ − a.e. (∀n ∈ ℕ) such that

π_{n}ϕ_{n} = r_{sp} (π_{n}) ϕ_{n} and ∥ϕ_{n}∥_{p} = 1.  (9.9)

Note that {ϕ_{n}; n ∈ ℕ} is sequentially relatively compact in σ (L^{p}, L^{p′}). Let λ be any limit point of the sequence {r_{sp} (π_{n}) , n → +∞} which is bounded by (ii). Then λ > 0 by (iii), and λ is the limit of a subsequence {r_{sp} (π_{n(k)}); k ∈ ℕ}. Take a sub-subsequence {m (l) ; l ∈ ℕ} of {n (k)} such that ϕ_{m(l)} → ϕ in σ (L^{p}, L^{p′}), as l → +∞. Then ϕ ≥ 0, μ-a.e.. We show now the key fact that ⟨ϕ⟩_{μ} > 0.

Indeed, if in contrary ⟨ϕ⟩_{μ} = 0, then ϕ_{m(l)} → 0 in probability μ. So, for any ε, δ > 0 fixed and for sufficiently large l, μ(ϕ_{m(l)} > δ) < ε, and then

1 = ∫ ϕ_{m(l)}^{p}dμ ≤ δ^{p} + sup_{μ(A)≤ε} ∫_{A} ϕ_{m(l)}^{p}dμ

≤ δ^{p} + \frac{1}{(r_{sp}(π_{m(l)}))^{p}} sup_{μ(A)≤ε} ∫_{A}(π_{m(l)}ϕ_{m(l)})^{p}dμ

≤ δ^{p} + (\frac{sup_{μ(A)≤ε} ∥1_{A}π_{m(l)}∥_{p,p}}{r_{sp}(π_{m(l)})} )^{p} .

Letting first ε → 0 and then δ → 0 we obtain by Lemma 3.1(a),

1 = lim_{n→∞} ∫ ϕ_{m(l)}^{p}dμ ≤ (\frac{∥π∥_{T (∞)}}{lim inf_{n→∞} r_{sp}(π_{n})})^{p}

which is in contradiction with assumption (iii).

We check now π_{∞}ϕ = λϕ. To that end note that for every g ∈ L^{∞},

|⟨π_{m(l)}ϕ_{m(l)} − π_{∞}ϕ, g⟩_{μ}|
= |⟨π_{m(l)}ϕ_{m(l)} − π_{∞}ϕ_{m(l)}, g⟩_{μ} + ⟨π_{∞}ϕ_{m(l)} − π_{∞}ϕ, g⟩_{μ}|
≤ |⟨ϕ_{m(l)}, (π_{m(l)}^{∗} − π_{∞}^{∗}) g⟩_{μ}| + |⟨ϕ_{m(l)} − ϕ, π_{∞}^{∗}g⟩_{μ}|
≤ ∥ (π_{m(l)}^{∗} − π_{∞}^{∗}) g∥_{p′} + |⟨ϕ_{m(l)} − ϕ, π_{∞}^{∗}g⟩μ|
⟶ 0

by the condition (i)′ and the convergence ϕ_{m(l)} → ϕ in σ(L^{p}, L^{p′}).

It follows from (9.9), as n = m(l) → ∞, that π_{∞}ϕ = λϕ. Hence λ ≤ r_{sp} (π_{∞}). Since λ is an arbitrary limit point of {r_{sp} (π_{n}) , n → +∞}, we obtain (9.8).

When π_{∞} is μ-essentially irreducible, by [55](the proof of Corollary 3.3) and π_{∞}ϕ = λϕ proved previously implies λ = r_{sp} (π_{∞}). The last claim follows.

Lemma 9.4. For a positive operator π on L^{p}(μ), and for 1 ≤ q < p < r ≤ +∞ with 1/p = θ/q + (1 − θ)/r,

∥π∥_{tail(L^{p})} ≤ (∥π∥_{tail(L^{q})})^{θ} · (∥π∥_{tail(L^{r})})^{1−θ}.  (9.10)

where ∥π∥_{tail(L^{∞})} := lim sup_{μ(A)→0} ∥1_{A}π∥_{∞}.

Proof. By Riesz-Thorin interpolation theorem, for any A ∈ B,

∥1_{A}π∥_{p} ≤ (∥1_{A}π∥_{q})^{θ} · (∥1_{A}π∥_{r})^{1−θ}

Now the desired result follows from Lemma 3.1 (and the definition above).

For the necessity in Theorem 3.9, we require the following general lower bound of large deviation.

Lemma 9.5. Assume that for some 0 < s ∈ 𝕋, P_{s} is μ-essentially irreducible (see (A2) for definition). Then for any measurable and τ-open subset G ⊂ M_{1}(E), for μ-a.s. x ∈ E,

lim inf_{t→∞} \frac{1}{t} log ℙ_{x}(L_{t} ∈ G) ≥ − inf_{β∈G} J_{μ}(β).

In particular for any non-empty initial measures A, if

∀B ∈ B_{μ}^{+}, ∃T > 0 : inf_{γ∈A} 𝔼^{ν}L_{T} (B) > 0  (9.11)

then

lim inf_{t→∞} \frac{1}{t} log inf_{γ∈A} ℙ_{γ}(L_{t} ∈ G) ≥ − inf_{β∈G} J_{μ}(β).  (9.12)

Proof. The first claim is [55], Theorem B.1. The second ”uniform lower bound” is established in the proof of Theorem 5.1 of [55] (the condition (9.11) is just (5.23) there).

9.2 Proof of the sufficiency of Theorem 3.9

We shall show (3.16) for A(p′, L) and claim that it is enough. Indeed, assumption (3.15) is satisfied for p greater, by Lemma 9.4 and the Markov property of P_{t_{0}}λR_{λ}. Hence when p is so close to infinity that p′ is smaller than 1 + ε, then (3.16) for A(p′, L) (∀L > 1) is stronger than that for A(1 + ε, L).

Fix then p ∈ (1, +∞) and fix A := A(p′, L) given in Theorem 3.9. By Lemma 9.2, we have only to establish the property (SBC) for Λ_{A}.

Step 1. For V ≤ δ and λ > δ, where λ, δ will be chosen later, let (P_{t}^{V}) is the Feynman-Kac semigroup defined in (9.2), and set

R_{λ}^{V} := ∑_{n=0}^{∞}(λ + 1)^{−n−1}(P_{1}^{V})^{n} or ∫_{0}^{∞} e^{−λt}P_{t}^{V} dt

according to 𝕋 = ℕ or ℝ^{+} (the associated resolvent). We have

P_{t_{0}}^{V} λR_{λ}^{V} ≤ e^{t_{0}δ} \frac{λ}{λ(δ)} · λ(δ)R_{λ(δ)}

where λ(δ) := (λ + 1)e^{−δ} − 1 if 𝕋 = ℕ, and λ(δ) := λ − δ if 𝕋 = ℝ^{+}. Denote the operator at its r.h.s. by π.

By assumption (3.15), we can choose λ > δ > 0 with λ(δ) = λ_{0} (λ_{0} being given in (3.15)) and with δ sufficiently small, so that

∥π∥_{tail} < 1.

Below we fix such λ > δ > 0.

Step 2. Let’s begin with the discrete time case: 𝕋 = ℕ. Obviously

Λ_{A}(V) ≤ lim_{n→∞} \frac{1}{n} log ∥(P^{V})^{n}∥_{p} = log r_{sp}(P^{V})  (9.13)

where P^{V} f = P_{1}^{V} f = e^{V} P f. By the spectral mapping theorem, for V ≤ δ (< λ),

r_{sp}(P_{t_{0}}^{V} λR_{λ}^{V}) = [r_{sp}(P^{V})]^{t_{0}}λ ∑_{n=0}^{∞}(λ + 1)^{−n−1}(r_{sp}(P^{V}))^{n}.  (9.14)

Given any sequence (V_{n}) ⊂ bB decreasing pointwise to zero on E with ∥V_{0}∥ ≤ δ, let π_{n} := P_{t_{0}}^{V_{n}}λR_{λ}^{V_{n}}. Since π_{n}1 ≥ 1, r_{sp}(π_{n}) ≥ 1. Moreover π_{n} ≤ π where π is constructed in Step 1 with ∥π∥_{tail} < 1. In further by dominated convergence, π_{n}f → P_{t_{0}}λR_{λ}f in L^{p}(μ) for each f ∈ L^{∞}.

Consequently all three conditions of Lemma 9.3 are satisfied and it gives us

lim sup_{n→∞} r_{sp}(P_{t_{0}}^{V_{n}}λR_{λ}^{V_{n}}) ≤ r_{sp}(P_{t_{0}}λR_{λ})

which equals to 1 by Lemma 3.2(a). Therefore by (9.14), r_{sp}(P^{V_{n}}) → 1 as n goes to infinity. Thus Λ_{A}(V_{n}) → 0 by (9.13), the desired (SBC).

Step 3. We now pass to the continuous time case: 𝕋 = ℝ^{+}. Note at first that

Λ_{A}(V) ≤ lim_{t→∞} \frac{1}{t} log ∥P_{t}^{V} 1∥_{p} =: Λ_{p}(V).  (9.15)

We have the following crucial inequality: for any |V | ≤ δ,

r_{sp}(P_{t_{0}}^{V} λR_{λ}^{V}) ≥ e^{t_{0}Λ_{p}(V)} · λ ∫_{0}^{∞} e^{−λt}e^{tΛ_{p}(V)}dt = \frac{exp[−t_{0}(λ − Λ_{p}(V))]}{λ − Λ_{p}(V)}  (9.16)

This was proved in (5.16) of [55] under the uniform integrability, but the argument in the proof of (5.16) in [55], relying only on (A1) and the existence of ground state, works still here because now the existence of ground state of P_{t_{0}}^{V} λR_{λ}^{V} is assured by Lemma 3.2(b). The detail is then omitted.

Now for any sequence (V_{n}) decreasing pointwise to zero on E with ∥V_{0}∥ ≤ δ, we see in a similar way that π_{n} := P_{t_{0}}^{V_{n}}λR_{λ}^{V_{n}} satisfies again the three conditions of Lemma 9.3 with π_{∞} = P_{t_{0}}λR_{λ}. Thus we obtain

lim sup_{n→∞} r_{sp}(P_{t_{0}}^{V_{n}}λR_{λ}^{V_{n}}) ≤ r_{sp}(P_{t_{0}}λR_{λ}) = 1.

Finally this, together with (9.16), implies Λ_{p}(V_{n}) → 0, where the desired (SBC) for Λ_{A} follows by (9.15).

□

9.3 Proof of the necessity of Theorem 3.9

(P_{t})_{t∈ℝ^{+}} is now symmetric and ergodic (by (A1)) w.r.t. the invariant measure μ. Thus P_{s} is ergodic w.r.t. μ for each s > 0 fixed by the spectral decomposition. Thus the lower bound of large deviations (9.12) in Lemma 9.5 holds for A := {γ} where γ is given in Theorem 3.9(b). Moreover we have by [55], Corollary B.11 (see Deuschel-Stroock [9], Theorem 4.2.58 for a partial result)

J_{μ}(β) = J_{E}(β) := \{ \begin{matrix}E(f, f), if β ≪ μ, f := (\frac{dβ}{dμ})^{1/2} ∈ 𝔻(E); \\ + ∞ otherwise \end{matrix}  (9.17)

Our goal to to prove (T5) in Theorem 3.5. To this end, let (f_{n})_{n∈ℕ} be a sequence satisfying ⟨f_{n}, f_{n}⟩ = 1 and ∥Lf_{n}∥_{2} → 0. Then by the contraction property of the Dirichlet form, E(|f_{n}|, |f_{n}|) ≤ E(f_{n}, f_{n}) = ⟨−Lf_{n}, f_{n}⟩ ⟶ 0. Let β_{n} := f_{n}^{2}dμ. By (9.17), J_{μ}(β_{n}) → 0.

Thus for any g ∈ bB and δ > 0, we have (by (9.12) in Lemma 9.5),

− inf\{J_{μ}(β); |β(g) − μ(g)| > δ\} ≤ lim inf_{t→∞} \frac{1}{t} log ℙ_{γ}(|L_{t}(g) − μ(g)| > δ) < 0

where the last inequality is our assumption of necessity. Consequently |β_{n}(g) − μ(g)| ≤ δ for n large enough. This means that |f_{n}|^{2} = f_{n}^{2} → 1 in the weak topology σ(L^{1}, L^{∞}), for g ∈ bB is arbitrary.

Finally by the well known Dunford-Pettis theorem (see [10], Vol.I, No.25, p.43), (f_{n}^{2})_{n∈ℕ} is uniformly integrable. Thus property (T5) holds and that implies the desired RTNC for (P_{t}) in L^{2}(μ) by Theorem 3.5, then in L^{p}(μ) for every 1 < p < +∞ by Lemma 9.4.

9.4 Proof of Theorem 3.8

Notice at first that by the Lumer-Phillips theorem ([60], Chap.IX, p250), ∥P_{t}∥_{2} ≤ 1 for all t ≥ 0 under the sector condition. Thus λR_{λ} are contractive on L^{2} for all λ > 0. Consequently (T5) is equivalent to (Tk), k=1,2,3,4 in Theorem 3.5.

(T1) ⟹ (T7): Fix some λ_{0} > 0 such that ∥λ_{0}R_{λ_{0}}∥_{tail} < 1. By Lemma 3.4, there is some r_{0} ≥ 0 and a non-increasing function β(r) on (r_{0}, +∞), such that

⟨f, f⟩ ≤ r∥(I − λ_{0}R_{λ_{0}})f∥_{2}^{2} + β(r)⟨|f|⟩^{2}.

The trick is: for any contraction operator C on L^{2},

2⟨(I − C)f, f⟩ − ⟨(I − C)f, (I − C)f⟩ = ⟨f, f⟩ − ⟨Cf, Cf⟩ ≥ 0, ∀f ∈ L^{2}.

Hence applying it to the contraction λ_{0}R_{λ_{0}}, we get by Lemma 8.1

∥(I − λ_{0}R_{λ_{0}})f∥_{2}^{2} ≤ 2⟨(I − λ_{0}R_{λ_{0}})f, f⟩ ≤ \frac{2κ^{2}}{λ_{0}} ⟨−Lf, f⟩_{μ}.

Subtituting it into the previous inequality, we obtain the weak Poincaré inequality of form type in (T7).

(T7) ⟹ (T6): (T7) implies, by Proposition 3.7, ∥P_{t}∥_{tail} ≤ exp(−t/r_{0}), which is much stronger than (T6).

(T7) ⟹ (T1): it is noted in Remarks (3.ii).

(T8) ⟹ (T5): easy for ∥Lf_{n}∥_{2} → 0 and ∥f_{n}∥_{2} ≤ 1 implies ⟨−Lf_{n}, f_{n}⟩_{μ} → 0.

(T7) ⟹ (T8) (under the sub-Markov property of (P_{t})): Let (E^{σ}, 𝔻(E^{σ})) be the closure of the nonnegative quadratic form E^{0}[f] := ⟨−Lf, f⟩ with 𝔻(E^{0}) = 𝔻_{2}(L) (the closability is assured by the sector condition, see [25]). Let −L^{σ} be the associated nonnegative self-adjoint operator (the symmetrization of −L), and let P_{t}^{σ} := e^{Lt}. (P_{t}^{σ}) is again a sub-Markov semigroup ([28]).

By (T7), (Q_{t} := exp(−t\sqrt{−L^{σ}})) satisfies the weak Poincaré inequality in (T4). Thus (T5) holds for (Q_{t}), i.e., for any sequence (f_{n}) ⊂ 𝔻_{2}(L) ⊂ 𝔻(E^{σ}) = 𝔻_{2}(\sqrt{−L^{σ}}) with ∥f_{n}∥_{2} ≤ 1 and E^{σ}[f_{n}] = ∥\sqrt{−L^{σ}}f_{n}∥_{2}^{2} → 0, \{f_{n}^{2}; n\} is uniformly integrable. This is exactly (T8).

The proof of Theorem 3.8 is completed. □

Of course if the symmetrization semigroup (P_{t}^{σ}) is again nonnegative, then (T 7) ⇒ (T 8) without the extra assumption that (P_{t}) is sub-Markov.

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Some open questions.

Q1. For a positive operator P on L^{1}(μ), if ∥P ∥_{tail} < r_{sp}(P), prove that r_{sp}(P) ∉ Σ_{ess}(P) (which is stronger than the spectral gap).

Q2. For the generator L = Δ − ∇V with Γ_{2} lower bounded, whether is-it true that on L^{2}(μ = e^{−V} dx),

λ_{ess−gap}(−L) = − lim sup_{N→∞} \frac{1}{N} log μ(\{x; ρ(x, o) > N\})?

Q3. For strongly Feller Markov semigroups which are topologically transitive on a Polish space E with invariant probability measure μ, whether is-it true that

inf{Reλ|λ ∈ Σ_{ess}(−L|_{L^{2}(μ)})}

= sup{λ > 0|∃K ⊂⊂ E, sup_{x∈K′} E^{x} exp(λτ_{K} ∘ θ_{1}) < +∞, ∀K ⊂ K′ ⊂⊂ E}?

or find important examples such that it is true.

Q4. In the context of Q.3, prove the inverse of Theorem 3.9 under reasonable conditions.

Q5. I believe that second quantification semigroup on the Poisson space given in Section 6.2 does not satisfy the RTNC if it has no spectral gap.