The domain of a generator and the intertwining property

Ichiro Shigekawa

1. Introduction

The generator of a semigroup is fundamental but it is rather difficult to give a precise characterization of the domain of the generator. Some generators are well-studied and we can give a precise domain of the generator. For example, the Laplacian on ℝ^{d} or the Ornstein-Uhlenbeck operator on the Wiener space in L^{2} setting are well-known.. In fact, the domain of the Laplacian on ℝ^{d} is the Sobolev space H^{2}(ℝ^{d}). Similar result holds for the Ornstein-Uhlenbeck operator.

We consider the semigroup {T_{t}} associated with the following type quadratic form on a measure space (X, B, m):

E(f, g) = ∫_{X}(∇f, ∇g)m(dx)

where ∇ is a gradient operator. Characterization of the form domain is rather easy. We will give a characterization of the generator domain. To do this, we assume the existence of another semigroup {T̂_{t}} satisfying the intertwining property:

∇T_{t} = T̂_{t}∇.

Under this condition, we will show that f belongs to the generator domain if and only if ∇f belongs to the form domain of Ê associated with {T̂_{t}}.

The organization of this paper is as follows. In Section 2, we give a precise characterization of the generator domain. The intertwining property plays an essential role. We can also give a sufficient condition for the intertwining property. In Section 3, we give some examples on ℝ^{d}. The generator domain involves the Bakry-Emery Γ_{2} (or equivalently the ‘Ricci’ curvature). Γ_{2} plays no role in some cases, but in other cases, it can not be neglected.

2. Intertwining semigroup and the generator domain

Let (X, B, m) be a σ-finite measure space. Suppose we are given a contraction symmetric semigroup {T_{t}} on L^{2}(m). We assume that all semigroups in the paper are strongly continuous and so in the sequel we do not mention it explicitly. We denote the generator by A and the associated quadratic form by E. We are interested in the domain of A, denoted by Dom(A). We regard Dom(A) as a Banach space, i.e., Dom(A) is equipped with the graph norm of A. In this paper, every subspace is equipped with its natural norm.

We intend to give a precise characterization of Dom(A). To do this, we suppose the following condition. We assume that the quadratic form E is of the following form:

E(f, g) = ∫_{X}(∇f, ∇g)_{K}m(dx).  (2.1)

Here, K is a Hilbert space, ∇ is a closed operator from L^{2}(m) into L^{2}(m; K) and L^{2}(m; K) is the set of all K-valued square integrable functions. The ∇ need not be a gradient operator. It is merely a closed operator. Suppose further that we are given another symmetric semigroup {T̂_{t}} on L^{2}(m; K). The generator is denoted by Â and the associated quadratic form by Ê. For a positive constant λ, we define Ê_{λ} by

Ê_{λ}(θ, η) = Ê(θ, η) + λ(θ, η)_{2},  (2.2)

where ( , )_{2} denotes the inner product in L^{2}.

Since Ê is bounded from below, we can choose a constant λ so that Ê_{λ} is strictly positive definite, i.e., there exists a positive constant c such that

Ê_{λ}(θ, θ) ≥ c(θ, θ)_{2}.

We take such λ and fix it.

We assume the following intertwining property of semigroups {T_{t}} and {T̂_{t}}: For any f ∈ Dom(∇), it holds that T_{t}f ∈ Dom(∇) and

∇T_{t}f = T̂_{t}∇f.  (2.3)

This property plays an important role in the paper. Since Dom(A) ⊆ Dom(∇), the above condition holds if and only if Dom(∇^{∗}) ⊆ Dom(Â) and

(Af, ∇^{∗}θ)_{2} = (∇f, Âθ)_{2}.  (2.4)

(see [5].)

We need the following proposition.

Proposition 2.1 We have Dom(Ê) ⊆ Dom(∇^{∗}). Moreover, there exists a constant C > 0 such that

∥∇^{∗}θ∥_{2} ≤ CÊ_{λ}(θ, θ)^{1/2}, ∀θ ∈ Dom(Ê)  (2.5)

where ∥ · ∥_{2} denotes the L^{2}-norm.

Proof. We first note that (λ − A)^{−1/2} can be expressed as

(λ − A)^{−1/2} = \frac{1}{Γ(1/2)} ∫_{0}^{∞} e^{−λt}t^{−1/2}T_{t}dt.

Hence, by the intertwining property (2.3), we have

∇(λ − A)^{−1/2}f = \frac{1}{Γ(1/2)} ∫_{0}^{∞} e^{−λt}t^{−1/2}∇T_{t}fdt

= \frac{1}{Γ(1/2)} ∫_{0}^{∞} e^{−λt}t^{−1/2}T̂_{t}∇fdt

= (λ − Â)^{−1/2}∇f.

This means that if f ∈ Dom(∇), then (λ − A)^{−1/2} ∈ Dom(∇) and it holds that

∇(λ − A)^{−1/2}f = (λ − Â)^{−1/2}∇f.  (2.6)

Define an operator S: L^{2}(m) → L^{2}(m; K) by S = ∇(λ − A)^{−1/2}. Clearly S is a bounded operator defined on the whole space L^{2}(m). In fact, this follows from the fact Dom(∇) = Dom(E).

Next we will show that for θ ∈ Dom(Ê), f ∈ Dom(E),

(θ, ∇f)_{2} = (S^{∗}(λ − Â)^{1/2}θ, f)_{2}.  (2.7)

To see this, we note

LHS = ((λ − Â)^{1/2}θ, (λ − Â)^{−1/2}∇f)_{2}
= ((λ − Â)^{1/2}θ, ∇(λ − A)^{−1/2}f)_{2} (∵ (2.6))
= ((λ − Â)^{1/2}θ, Sf)_{2}
= (S^{∗}(λ − Â)^{1/2}θ, f)_{2}
= RHS.

(2.7) implies that θ ∈ Dom(∇^{∗}) and

∇^{∗}θ = S^{∗}(λ − Â)^{1/2}θ.  (2.8)

We have therefore

∥∇^{∗}θ∥_{2} ≤ ∥S^{∗}∥_{op} ∥(λ − Â)^{1/2}θ∥_{2} = ∥S^{∗}∥_{op} Ê_{λ}(θ, θ)^{1/2},

where ∥ ∥_{op} denotes the operator norm. Now the proof is complete. □

Now we can show the following main theorem in this section.

Theorem 2.2 f ∈ Dom(A) if and only if ∇f ∈ Dom(Ê). We have moreover

(Af, Af)_{2} = Ê(∇f, ∇f), ∀f ∈ Dom(A).  (2.9)

Proof. Take any f ∈ Dom(A) and fix it. We define Φ by

Φ(θ) = −(Af, ∇^{∗}θ) + λ(∇f, θ)_{2}, θ ∈ Dom(Ê).

Φ(θ) is well-defined by Proposition 2.1. Moreover we have

|Φ(θ)| ≤ ∥Af∥_{2}∥∇^{∗}θ∥_{2} + λ∥∇f∥_{2}∥θ∥_{2}
≤ C∥Af∥_{2}Ê_{λ}(θ, θ)^{1/2} + λ∥∇f∥_{2}∥θ∥_{2}

which deduces that Φ is a bounded linear functional on Dom(Ê). Here the topology in Dom(Ê) is defined by the inner product Ê_{λ}. Hence there exists η ∈ Dom(Ê) such that

Φ(θ) = Ê_{λ}(η, θ).

On the other hand, if θ ∈ Dom(Â), then it holds that

(Af, ∇^{∗}θ)_{2} = (∇f, Âθ)_{2}.

Hence we have for θ ∈ Dom(Â),

(η, (λ − Â)θ)_{2} = Ê_{λ}(η, θ)
= Φ(θ)
= −(Af, ∇^{∗}θ)_{2} + λ(∇f, θ)_{2}
= −(∇f, Âθ)_{2} + λ(∇f, θ)_{2}
= (∇f, (λ − Â)θ)_{2}.

Since (λ − Â)θ runs over the whole space L^{2}(m; K), we have ∇f = η. Thus we have obtained that ∇f ∈ Dom(Ê).

Conversely, we assume that ∇f ∈ Dom(Ê). Then we have that ∇f ∈ Dom(∇^{∗}) by Proposition 2.1. Since A = −∇^{∗}∇, we have f ∈ Dom(A).

Lastly let us show (2.9). For f ∈ Dom(A), θ ∈ Dom(Â), it holds that

(Af, ∇^{∗}θ)_{2} = (∇f, Âθ)_{2} = −Ê(∇f, θ).  (2.10)

We note that Dom(A) is dense in Dom(Ê) and ∇^{∗} is continuous in Dom(Ê). Hence, by taking limit in (2.10), we have

(Af, ∇^{∗}θ)_{2} = −Ê(∇f, θ), ∀θ ∈ Dom(Ê).

This leads, by setting θ = ∇f,

(Af, ∇^{∗}∇f)_{2} = −Ê(∇f, ∇f).

Now we have (2.9) since ∇^{∗}∇ = −A. □

Making use of the above theorem, we can give a sufficient condition for the intertwining property.

Proposition 2.3 Assume the following two conditions:

(i) Dom(Ê) ⊆ Dom(∇^{∗}) and there exists a constant C > 0 such that

∥∇^{∗}θ∥_{2} ≤ CÊ_{λ}(θ, θ)^{1/2}, ∀θ ∈ Dom(Ê).

(ii) There exist a subspace D that is dense in Dom(A) and a subspace D̂ that is dense in Dom(Ê) such that ∇D ⊆ Dom(Ê) and

(Af, ∇^{∗}θ)_{2} = −Ê(∇f, θ), ∀f ∈ D, ∀θ ∈ D̂.  (2.11)

Then the intertwining property (2.4) holds.

Proof. By the condition (i), (2.11) holds for f ∈ D, θ ∈ Dom(Ê). Since ∇f ∈ Dom(Ê) for f ∈ D, we can set θ = ∇f and have

(Af, ∇^{∗}∇f)_{2} = −Ê(∇f, ∇f), ∀f ∈ D.

Hence we have

(Af, Af)_{2} + λ(∇f, ∇f)_{2} = Ê_{λ}(∇f, ∇f), ∀f ∈ D.  (2.12)

Now for any f ∈ Dom(A), take a sequence {f_{n}} ⊆ D such that f_{n} converges to f in Dom(A). Then, (2.12) implies that {∇f_{n}} is a Cauchy sequence in Dom(Ê). This, combined with the fact that ∇f_{n} → ∇f in L^{2}(m; K), implies ∇f ∈ Dom(Ê). Moreover, by taking limit, (2.11) holds for f ∈ Dom(A) and θ ∈ Dom(Ê). If, in particular, θ ∈ Dom(Â), then we have

(Af, ∇^{∗}θ)_{2} = −Ê(∇f, θ) = (∇f, Âθ)_{2}

which completes the proof. □

The next corollary is easy.

Corollary 2.4 Assume the following two conditions:

(i) Dom(Ê) ⊆ Dom(∇^{∗}) and there exists a constant C > 0 such that

∥∇^{∗}θ∥_{2} ≤ CÊ_{λ}(θ, θ)^{1/2}, ∀θ ∈ Dom(Ê).

(ii) There exist a subspace D that is dense in Dom(A) such that

∇Af = Â∇f, ∀f ∈ D.  (2.13)

(AD ⊆ Dom(∇) and ∇D ⊆ Dom(Â) are assumed).

Then the intertwining property (2.4) holds.

Proof. From the condition (ii), we have for θ ∈ Dom(Ê),

(Af, ∇^{∗}θ)_{2} = (∇Af, θ)_{2} = (Â∇f, θ)_{2} = −Ê(∇f, θ).

Now the rest is easy. □

3. Examples

In this section, we give examples on ℝ^{d}. We define a measure m on ℝ^{d} by m(dx) = e^{−U(x)}dx where U is a C^{∞} function. We consider the following Dirichlet form in L^{2}(m).

E(f, g) = ∫_{ℝ^{d}}(∇f(x), ∇g(x))m(dx).  (3.1)

Here, of course, ∇ is the gradient operator: ∇f = (∂_{1}f, ∂_{2}f, …, ∂_{d}f), ∂_{j} = \frac{∂}{∂x^{j}}. The associated generator A is

Af = ∑_{j}(∂_{j}∂_{j}f − ∂_{j}U∂_{j}f).

Since ∇f is 1-form, we need a semigroup acting on 1-forms. Moreover, this semigroup must satisfy the intertwining property. The semigroup is generated by the Hodge-Kodaira Laplacian −(dd^{∗} + d^{∗}d), where d is the exterior differentiation and d^{∗} is its dual operator in L^{2}(m; ℝ^{d}). The following formula is well-known as the Weitzenböck formula:

dd^{∗} + d^{∗}d = ∇^{∗}∇ + R.  (3.2)

Here ∇ is the covariant differentiation and R is given as

R(∑_{j} θ_{j}dx^{j}) = ∑_{j,k} ∂_{j}∂_{k}Uθ_{j}dx^{k}.

Note that (dd^{∗} + d^{∗}d)d = d(dd^{∗} + d^{∗}d) on C_{0}^{∞}(ℝ^{d}). Here C_{0}^{∞} stand for the set of all C^{∞} functions with compact support. Then, by virtue of Corollary 2.4, the intertwining property holds for A = −∇^{∗}∇ and Â = −(dd^{∗}+d^{∗}d). The quadratic form associated with Â is given by

Ê(θ, η) = (dθ, dη)_{2} + (d^{∗}θ, d^{∗}η)_{2} = (∇θ, ∇η)_{2} + (Rθ, η)_{2}  (3.3)

We decompose Ê into two parts. To do this, define E⃗ by

E⃗(θ, η) = (∇θ, ∇η)_{2}

and Q_{R} by

Q_{R}(θ, η) = (Rθ, η)_{2}.  (3.4)

From now on, we assume that R is bounded from below: there exists a constant γ such that R(x) ≥ γI for all x ∈ ℝ^{d}. Here this means that the matrix R(x) − γI is non-negative definite.

Proposition 3.1 Assume that R is bounded from below. Then it holds that

Dom(Ê) = Dom(E⃗) ∩ Dom(Q_{R}).  (3.5)

Proof. We first remark that (3.5) is not trivial. In fact, Ê is a form closure of (Ê, C_{0}^{∞}(ℝ^{d} → ℝ^{d})). So Dom(Ê) ⊆ Dom(E⃗) ∩ Dom(Q_{R}) is clear.

To show the converse, we prove that C_{0}^{∞}(ℝ^{d} → ℝ^{d}) is a form core for E⃗ ∔ Q_{R}, where E⃗ ∔ Q_{R} stands for the form sum.

Take any θ ∈ Dom(E⃗) ∩ Dom(Q_{R}). Define a function ψ_{ε}: ℝ → ℝ by

ψ_{ε}(t) = \frac{1}{1 + εt}.

Then

|∇(ψ_{ε}(|θ|)θ)| ≤ ψ_{ε}(|θ|)|∇θ| + |θ|ψ′_{ε}(|θ|)|∇θ|.

Hence E⃗(ψ_{ε}(|θ|)θ, ψ_{ε}(|θ|)θ) is uniformly bounded as ε → 0. By taking a suitable sequence {ε_{n}} which converges to 0, the Cesàro mean of {ψ_{ε_{n}}(|θ|)θ} converges in Dom(E⃗). Since ψ_{ε}(|θ|)θ → θ in L^{2} as ε → 0, the limit is equal to θ. Moreover, by the dominated convergence theorem, it converges in Dom(Q_{R}). Thus we have seen that any θ ∈ Dom(E⃗) ∩ Dom(Q_{R}) can be approximated by functions in Dom(E⃗)∩Dom(Q_{R})∩L^{∞}.

Next, take θ ∈ Dom(E⃗)∩Dom(Q_{R})∩L^{∞}. We introduce a non-negative function ϕ ∈ C^{∞}(ℝ) such that

ϕ(t) = \{ \begin{matrix}1, & t ≤ 0, \\ 0, & t ≥ 1. \end{matrix}

Set θ_{n}(x) = θ(x)ϕ(|x| − n), n ∈ ℕ. Then it is not difficult to see that ∇θ_{n} converges to ∇θ in L^{2}. This means that θ_{n} converges to θ in Dom(E⃗). The convergence in Dom(Q_{R}) is also easy.

Lastly, by using molifier, any θ ∈ Dom(E⃗) ∩ Dom(Q_{R}) ∩ L^{∞} with compact support can be approximated by functions in C_{0}^{∞}(ℝ^{d} → ℝ^{d}). This completes the proof. □

Now the following theorem is an easy consequence of Theorem 2.2 and Proposition 3.1.

Theorem 3.2 Assume that R is bounded from below. Then f ∈ Dom(A) if and only if f ∈ Dom(∇^{2}) ∩ Dom(Q_{R}).

Here Dom(∇^{2}) is the set of all functions f with f, ∇f, ∇^{2}f ∈ L^{2}. If R is not large, we can delete the restriction Dom(Q_{R}) as follows.

Proposition 3.3 Assume that there exists γ > 0 such that ∇^{2}U(x) ≥ γI for large x. Further we assume that there exists ε > 0 such that e^{ε∥∇^{2}U∥_{op}} ∈ L^{1}(m). Then f ∈ Dom(A) if and only if f ∈ Dom(∇^{2}).

Proof. It is enough to prove that Dom(E⃗) ⊆ Dom(Q_{R}). From the assumptions, we can find V ∈ C_{0}^{∞}(ℝ^{d}) and a constant γ′ > 0 such that

∇^{2}(U + V)(x) ≥ γ′I, ∀x ∈ ℝ^{d}.

Now Bakry-Emery criterion ([1]) implies the logarithmic Sobolev inequality for the Dirichlet form with the reference measure e^{−U−V} dx. Since m has a bounded density with respect to the measure e^{−U−V} dx, the logarithmic Sobolev inequality holds for E (see Holley-Stroock [3]). Thus there exists a constant α such that

∫_{ℝ^{d}} |θ(x)|^{2} log(|θ(x)|/∥θ∥_{2})dx ≤ αE⃗(θ, θ).

By the Hausdorff-Young inequality st ≤ s log s − s + e^{t}, we have

(R(x)θ(x), θ(x)) ≤ ∥R(x)∥_{op}|θ(x)|^{2}

≤ (|θ(x)|^{2}/ε) log(|θ(x)|^{2}/ε) − (|θ(x)|^{2}/ε) + e^{ε∥R(x)∥_{op}}.

The right hand side is integrable for θ ∈ Dom(E⃗) and hence θ ∈ Dom(Q_{R}). This completes the proof. □

On the contrary, there exists an example that Q_{R} is essential. We consider the case d = 1 and set U(x) = e^{x^{2}}. Then U′′(x) = e^{x^{2}}(4x^{2} + 2).

We assume that there exists a constant K > 0 such that for any f ∈ C^{2}(ℝ)

∫_{ℝ}(f′)^{2}e^{x^{2}}(4x^{2} + 2)e^{−U(x)}dx ≤ K{∫_{ℝ}(f^{2} + (f′)^{2} + (f′′)^{2})e^{−U(x)}dx}. (3.6)

Take function f_{v} such that f′′_{v}(x) = \frac{1}{\sqrt{2πv}}e^{−(x−n)^{2}/2v}. By taking limit v → 0, we have f_{v} → (x − n)_{+}, f′_{v} → 1_{[n,∞)} and (f′′_{v})^{2} → δ_{n}/\sqrt{2}. Here (x − n)_{+} = (x − n) ∨ 0 = max{x − n, 0} and δ_{n} is the Dirac measure at n. Hence we have from (3.6),

∫_{n}^{∞} e^{x^{2}}(4x^{2} + 2)e^{−U(x)}dx
≤ K\{∫_{n}^{∞}((x − n)_{+}^{2} + 1)e^{−U(x)}dx + e^{−U(n)}/\sqrt{2}\}. (3.7)

But

∫_{n}^{∞} e^{x^{2}}(4x^{2} + 2)e^{−U(x)}dx ≥ e^{n^{2}} ∫_{n}^{∞}((x − n)_{+}^{2} + 1)e^{−U(x)}dx

and

∫_{n}^{∞} e^{x^{2}}(4x^{2} + 2)e^{−U(x)}dx ≥ 2n ∫_{n}^{∞} 2e^{x^{2}}xe^{−U(x)}dx

≥ 2n ∫_{n}^{∞} −\frac{d}{dx}e^{−U(x)}dx

= 2ne^{−U(n)}.

This contradicts (3.7). Thus Dom(A) ⊊ Dom(∇^{2}).

Lastly, we give another example such that Dom(A) ⊊ Dom(∇^{2}). This time, we discuss the infinite dimensional space ℝ^{ℕ}. For n ∈ ℕ, we define a measure on ℝ by

μ_{n}(dx) = \frac{1}{\sqrt{2π/n}}e^{−nx^{2}/2}.

Let μ = μ_{1} × μ_{2} × · · ·. We define a Dirichlet form in L^{2}(μ) by

E(f, g) = ∫_{ℝ^{ℕ}}(∇f, ∇g)μ(dx).

Here ∇f(x) = (∂_{1}f(x), ∂_{2}f(x), …). The generator L is given by

Lf(x) = ∑_{n}(\frac{∂^{2}f}{∂^{2}x^{n}} (x) − nx^{n} \frac{∂f}{∂x^{n}} (x)).

This is well-defined for f ∈ FC_{0}^{∞}(ℝ^{ℕ}) and L is essentially selfadjoint on FC_{0}^{∞}(ℝ^{ℕ}). Here FC_{0}^{∞}(ℝ^{ℕ}) is the set of all functions of the form

f(x) = F (x^{1}, x^{2}, ⋯, x^{n}), n ∈ ℕ, F ∈ C_{0}^{∞}(ℝ^{n}).

In this case, we can show that Dom(L) ⊊ Dom(∇^{2}). To see this, set f_{n}(x) = x^{n}. Then we have ∥f_{n}∥_{2}^{2} = \frac{1}{n}, ∥∇f_{n}∥_{2}^{2} = 1 and ∇^{2}f_{n} = 0. On the other hand, Lf_{n} = −nf_{n}, and hence ∥Lf_{n}∥_{2}^{2} = n. Therefore we have the desired result.

References

[1] D. Bakry and M. Emery. Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris, 299 (1984), 775–778.

[2] J-D. Deuschel and D. W. Stroock. “Large deviations,” Academic Press, San Diego, 1989.

[3] R. Holley and D. W. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models, J. Statist. Phys., 46 (1987), 1159–1194.

[4] I. Shigekawa. L^{p} contraction semigroups for vector valued functions, J. Funct. Anal., 147 (1997), 69–108.

[5] I. Shigekawa. Semigroup domination on a Riemannian manifold with boundary, preprint.

Ichiro Shigekawa

Department of Mathematics, Graduate School of Sciences,

Kyoto University, Kyoto 606-8502, Japan