A NON-SYMMETRIC DIFFUSION PROCESS ON THE WIENER SPACE

ICHIRO SHIGEKAWA

Abstract. We discuss a non-symmetric diffusion process on the Wiener space. The process we consider is generated by A = L + b, L being the Ornstein-Uhlenbeck operator and b being a vector field. Under suitable integrability condition for b, we show the existence of associated diffusion process. We also investigate the domain of the generator. Further we consider a similar problem in the finite dimensional Euclidean space.

1. Introduction

We construct a non-symmetric diffusion process on an abstract Wiener space (B, H, μ). Here B is a real separable Banach space, μ is a Gaussian measure with a reproducing kernel Hilbert space H. The generator of the diffusion process we consider is of the form A = L + b, where L is the Ornstein-Uhlenbeck operator and b is an H-valued function. We regard b as a first order differential operator and so our generator A is, using the H-derivative D, given by

Af = Lf +_{H}⟨b, Df⟩_{H^{∗}}.  (1.1)

We assume the following condition:

(A.1) exp{|b|^{2}} ∈ L^{2+}(μ) := ⋃_{p>2} L^{p}(μ).

We construct a diffusion process in the framework of Dirichlet form (to be precise, semi-Dirichlet form). In our case, the bilinear form E is given by

E(f, g) = (−Af, g) = ∫_{B}{(Df, Dg)_{H^{∗}} − ⟨Df, gb⟩}dμ.  (1.2)

To show the existence, we appeal to the general theory of Ma-Overbeck-Röckner [4] which ensure the existence of diffusion processes associated with non-symmetric semi-Dirichlet forms. We also discuss an issue of the domain of the generator. Further we consider a similar problem in the finite dimensional Euclidean space.

The organization of this paper is as follows. We review non-symmetric diffusion processes in the section 2. We recall conditions to ensure the existence of diffusion processes. We will check all conditions for our diffusion process. A main point is to see the equivalence between our bilinear form and that of the Ornstein-Uhlenbeck process. In the section 3, we determine the domain of the generator. We invoke the perturbation theorem and see that the domain coincides with the domain of the Ornstein-Uhlenbeck operator. Our main tool is the logarithmic Sobolev inequality. In the finite dimensional case, we can make use of the Sobolev inequality and so we consider similar problem in the finite dimensional Euclidean space in the section 4.

1991 Mathematics Subject Classification. 60J60, 60H07.

2. Non-symmetric diffusion

We first recall necessary notions from Ma-Overbeck-Röckner [4] which ensure the existence of diffusion processes.

Since our space is a separable Banach space, it is a good space as a topological space, i.e, a Polish space. We need to check the following conditions:

(1) There exists a constant γ ≥ 0 so that Ẽ_{γ} = Ẽ+γ(· , ·)_{2} is positive definite and closed. Here ( , )_{2} is the inner product in L^{2}(μ) and Ẽ is the symmetric part of E:

Ẽ(f, g) = \frac{1}{2}\{E(f, g) + E(g, f)\}.

(2) There exists a constant K such that

|E(f, g)| ≤ KẼ_{γ}(f, f)^{1/2}Ẽ_{γ}(g, g)^{1/2}.

(3) If f ∈ Dom(E), then f^{+} ∧ 1 ∈ Dom(E) and

E(f + f^{+} ∧ 1, f − f^{+} ∧ 1) ≥ 0.

(4) Each element of Dom(E) admits a quasi-continuous version.

(5) E_{γ}-capacity is tight. Even though the case γ = 0 was treated in [4], there is no essential difference.

We now turn to our case A = L+b. The associated bilinear form E is given by (1.2). We use a perturbation method. Our basic generator is the Ornstein-Uhlenbeck operator and so we need several facts for the Ornstein-Uhlenbeck operator L. We denote the bilinear form associated with the Ornstein-Uhlenbeck operator by E^{O-U}, i.e.,

E^{O-U}(f, g) = ∫_{B}(Df, Dg)_{H^{∗}}dμ.  (2.1)

Our main tool is the following logarithmic Sobolev inequality (due to Gross [2]): for f ∈ Dom(E^{O-U}),

∫_{B} |f|^{2} log{|f|/∥f∥_{2}}dμ ≤ ∫_{B} |Df|^{2}dμ.  (2.2)

Here ∥·∥_{2} denotes the L^{2} norm. When p ≥ 2, by substituting f by |f|^{p/2} sgn(f), we easily have

∫_{B} |f|^{p} log\{|f|/∥f∥_{p}\}dμ ≤ \frac{p}{2} ∫_{B} |Df|^{2}|f|^{p−2}dμ.  (2.3)

We use this inequality later to show that A is dissipative in L^{p}. Now let us check the first condition (1). First of all, we have to give the domain of E. By the following proposition, E is well-defined on Dom(E^{O-U}).

Proposition 2.1. There exist γ > 0 and C > 0 such that

E(f, f) ≤ C{E^{O-U}(f, f) + (f, f)_{2}}  (2.4)
C^{−1}E^{O-U}(f, f) ≤ E(f, f) + γ(f, f)_{2}.  (2.5)

Proof. First we note

|⟨fb, Df⟩| ≤ \frac{1}{2}\{|Df|^{2} + |f|^{2}|b|^{2}\}.

To estimate the second term in the right hand side, we recall the Hausdorff-Young inequality st ≤ s log s − s + e^{t}. Using this, we have

|f|^{2}|b|^{2} ≤ \frac{|f|^{2}}{2(1 + ε)} log \frac{|f|^{2}}{2(1 + ε)} − \frac{|f|^{2}}{2(1 + ε)} + e^{2(1+ε)|b|^{2}}.

When ∥f∥_{2} = 1, we have

∫_{B} |f|^{2}|b|^{2}dμ ≤ \frac{1}{1 + ε} ∫_{B} |f|^{2} log |f|dμ − \frac{1}{2(1 + ε)} ∫_{B} |f|^{2} log 2(1 + ε)dμ

− ∫_{B} \frac{|f|^{2}}{2(1 + ε)}dμ + ∥e^{2(1+ε)|b|^{2}}∥_{1}

≤ \frac{1}{1 + ε} ∫_{B}(Df, Df)dμ − \frac{1}{2(1 + ε)}\{1 + log 2(1 + ε)\} + ∥e^{2(1+ε)|b|^{2}}∥_{1}.

Hence, for general f,

∫_{B} |f|^{2}|b|^{2}dμ ≤ \frac{1}{1 + ε} ∫_{B}(Df, Df)dμ + \{∥e^{2(1+ε)|b|^{2}}∥_{1} − \frac{1}{2(1 + ε)} log 2e(1 + ε)\}∥f∥_{2}^{2}

≤ \frac{1}{1 + ε} ∫_{B}(Df, Df)dμ + 2γ∥f∥_{2}^{2}

where we set

2γ = ∥e^{2(1+ε)|b|^{2}}∥_{1} − \frac{1}{2(1 + ε)} log 2e(1 + ε).

Thus we have

E(f, f) ≥ ∫_{B} |Df|^{2}dμ − \frac{1}{2} ∫_{B}(|Df|^{2} + |f|^{2}|b|^{2})dμ

≥ ∫_{B} |Df|^{2}dμ − \frac{1}{2} ∫_{B} |Df|^{2}dμ − \frac{1}{2(1 + ε)} ∫_{B} |Df|^{2}dμ − γ∥f∥_{2}^{2}

≥ \frac{ε}{2(1 + ε)} ∫_{B} |Df|^{2}dμ − γ∥f∥_{2}^{2}.

From this

E(f, f) + γ∥f∥_{2}^{2} ≥ \frac{ε}{2(1 + ε)} ∫_{B} |Df|^{2}dμ.  (2.6)

Similarly we have

E(f, f) ≤ \frac{4 + 3ε}{2(1 + ε)} ∫_{B} |Df|^{2}dμ + γ∥f∥_{2}^{2}

which shows (2.4).

By the above proposition, Ẽ is equivalent with the following form

E1^{O-U}(f, g) = ∫_{B}(Df, Dg)_{H^{∗}}dμ + (f, g)_{L^{2}},

but it is well known that this form is closed. Now we have shown that Ẽ_{γ} with the domain Dom(E^{O-U}) is positive definite and closed. This shows (1).

(2) is shown in the similar manner.

Instead of (3), we will show the following:

E(f^{+} ∧ 1, f − f^{+} ∧ 1) ≥ 0.

(Equivalence can be seen in [5, Theorem I.4.4].) Moreover it is enough to show the following (see, e.g., [5, Proposition I.4.7]): for any ε > 0 there exists φ_{ε}: ℝ → [−ε, 1 + ε] such that φ_{ε}(t) = t for t ∈ [0, 1], 0 ≤ φ_{ε}(t_{2}) − φ_{ε}(t_{1}) ≤ t_{2} − t_{1} if t_{1} ≤ t_{2}, φ_{ε}(f) ∈ Dom(E) and

lim inf_{ε→0} E(φ_{ε}(f), f − φ_{ε}(f)) ≥ 0.

We take any smooth function φ_{ε} such that

φ_{ε}(t) = \{ \begin{matrix}−ε, & t ≤ −2ε, \\ 1 + ε, & t ≥ 1 + 2ε. \end{matrix}

Now

E(φ_{ε}(f), f − φ_{ε}(f)) = ∫_{B}(Dφ_{ε}(f), D(f − φ_{ε}(f)))_{H^{∗}}dμ − ∫_{B}(Dφ_{ε}(f), b(f − φ_{ε}(f)))_{H^{∗}}dμ

= ∫_{B} φ′_{ε}(f)(1 − φ′_{ε}(f))|Df|_{H^{∗}}^{2}dμ − ∫_{B} φ′_{ε}(f)(Df, b)_{H^{∗}}(f − φ_{ε}(f))dμ.

On the other hand,

∫_{B} |φ′_{ε}(f)(Df, b)_{H^{∗}}(f − φ_{ε}(f))|dμ ≤ ∫_{{−2ε≤f≤0}∪{1≤f≤1+2ε}} |Df||b||f|dμ.

Hence

lim sup_{ε→0} ∫_{B} |φ′_{ε}(f)(Df, b)_{H^{∗}}(f − φ_{ε}(f))|dμ ≤ ∫_{{f=0}∪{f=1}} |Df||b||f|dμ = 0.

Here we used that for any constant c,

∫_{{f=c}} |Df|dμ = 0.

Thus we have

lim inf_{ε→0} E(φ_{ε}(f), f − φ_{ε}(f)) ≥ 0

as desired.

(4) can be shown by a standard method because Dom(E) ∩ C(B) is dense in Dom(E). Lastly we show (5), the tightness of the capacity. Before proving this let us review the

capacity. In the following, we take α so that Ẽ_{α} is equivalent with E^{O-U}, i.e., Proposition 2.1 holds for γ = α. Hence Ẽ_{α} is positive definite and the resolvent G_{α} exists. For any function h on B and an open set U, set

L_{h,U} = {w ∈ Dom(E) |w ≥ h a.e. on U}.  (2.7)

Then it holds that (see Ma-Röckner [5, Chapter III.1.])

i) there exists a unique h_{U} such that

E_{α}(h_{U}, w) ≥ E_{α}(h_{U}, h_{U}) ∀w ∈ L_{h,U}.  (2.8)

ii) h_{U} is α-excessive and if w ∈ Dom(E) and w ≥ 0 a.e. on U, then E_{α}(h_{U}, w) ≥ 0. iii) If, in addition, h is α-excessive, then

0 ≤ h_{U} ≤ h, h_{U} = h on U.

Using this notation, the capacity is defined as follows. We take any α-excessive functions h, and co-α-excessive ĝ and fix them. For any open set U, set

Cap_{h,g}(U) := E_{α}(h_{U}, ĝ_{U}).  (2.9)

In particular, take any ϕ ∈ L^{2} such that ϕ > 0 a.e., and set h = G_{α}ϕ, ĝ = Ĝ_{α}ϕ. Then,

Cap_{h,g}(U) = (h_{U}, ϕ) = (ϕ, ĝ_{U}) = E_{α}(h_{U}, ĝ).  (2.10)

Here, we used that ĝ_{U} = ĝ a.e. on U.

In the following we take ϕ = 1 and we write Cap in place of Cap_{h,g}. Since A1 = 0, we have αG_{α}1 = 1, i.e., h = 1/α. Since the Ornstein-Uhlenbeck operator is self-adjoint, the associated capacity is characterized as follows:

Cap^{O-U}(U) = inf{E1^{O-U}(w, w) ; w ≥ 1 a.e. on U.}  (2.11)

This capacity can be extended for arbitrary sets as an outer capacity.

Proposition 2.2. There exists C > 0 such that for any set E,

Cap(E)^{2} ≤ C Cap^{O-U}(E)  (2.12)

Proof. It is enough to show this for an open set U. First we note that

Cap(U) = E(h_{U}, ĝ)
≤ KE_{α}(h_{U}, h_{U})^{1/2}E_{α}(ĝ, ĝ)^{1/2}.

Further, for any w ∈ L_{h,U}, i.e., w ≥ 1/α a.e. on U,

E_{α}(h_{U}, h_{U}) ≤ E_{α}(h_{U}, w)
≤ KE_{α}(h_{U}, h_{U})^{1/2}E_{α}(w, w)^{1/2}

which implies

E_{α}(h_{U}, h_{U}) ≤ K^{2}E_{α}(w, w)
≤ K_{1} ∫_{B}(|Dw|^{2} + |w|^{2})dμ.

Thus we have

Cap(U)^{2} ≤ K^{2}K_{1}E_{α}(ĝ, ĝ) inf{∫_{B}(|Dw|^{2} + |w|^{2})dμ; w ≥ 1/α a.e. on U}. (2.13)

This means that Cap is dominated by the capacity associated with the Ornstein-Uhlenbeck process up to constant.

It is well-known that the capacity associated with the Ornstein-Uhlenbeck process is tight. Now the tightness of our capacity follows.

The above proposition is sufficient to show the tightness. But one may ask whether the reversed estimate holds and we have

Proposition 2.3. Cap and Cap^{O-U} have the common capacity zero sets, i.e., Cap(V) = 0 if and only if Cap^{O-U}(V) = 0.

Proof. We use a characterization of capacity zero sets by means of Dirichlet form. An increasing sequence of closed set {F_{n}} is called a nest if ⋃_{n} Dom(E)_{F_{n}} is dense in Dom(E). Here, Dom(E)_{F_{n}} is a set of all elements in Dom E with support in F_{n}. Then, a set N is of zero capacity if and only if there exists a nest {F_{n}} such that N ⊆ ⋂_{n} F_{n}^{c}.

Now the result follows easily since E and E^{O-U} have the same domain and the same topology.

If we impose an additional condition, we can give a direct estimate between two capacities. To show this, we recall that the reduced function is defined for an arbitrary set by using the quasi-continuous function. For any set V, put

L_{h,V} = {w ∈ Dom(E) |w̃ ≥ h Cap-q.e. on V }.  (2.14)

Here, w̃ denotes the quasi-continuous modification of w with respect to the capacity Cap. We can show that there exists h_{V} satisfying (2.8) as well. Moreover similar properties hold for h_{V} if we replace a.e. with q.e. We denote the reduced function with respect to E1^{O-U} by e_{V}. Further we assume that ĝ is Cap^{O-U}-quasi-continuous and hence Cap-quasi continuous by Proposition 2.2.

Proposition 2.4. There exists a constant C > 0 such that for any λ > 0 and any set U,

Cap^{O-U}(U ∩ {ĝ > λ}) ≤ Cλ^{−1} Cap(U ∩ {ĝ > λ})  (2.15)

Proof. We recall that for any set V,

Cap(V) = E(h_{V} , ĝ),
Cap^{O-U}(V) = E^{O-U}(e_{V} , e_{V}).

Setting V_{λ} = U ∩ {ĝ > λ}, we have

Cap^{O-U}(V_{λ}) = E1^{O-U}(e_{V_{λ}}, e_{V_{λ}})

≤ E1^{O-U}(h_{V_{λ}}, h_{V_{λ}}) (∵ h_{V_{λ}} ≥ 1 q.e. on V_{λ} and e_{V_{λ}} is the minimizing element)

≤ C_{1}E_{α}(h_{V_{λ}}, h_{V_{λ}}) (∵ Proposition 2.1)
≤ C_{1}λ^{−1}E_{α}(h_{V_{λ}}, ĝ) (∵ λ^{−1}ĝ ≥ 1 on V_{λ})
= C_{1}λ^{−1} Cap(V_{λ})

which completes the proof.

3. Domain of the generator

In this section, we determine the domain of the generator.

Let {T_{t}} be the semigroup associated with the semi-Dirichlet form E. There exists γ ≥ 0 such that {e^{−γt}T_{t}} is a contraction semigroup in L^{2}. By the interpolation, {T_{t}} is also a strongly continuous contraction semigroup in L^{p}. To specify the space, we denote the semigroup by {T_{t}^{(p)}} and its generator by A_{(p)}.

As before let E^{O-U} be the Dirichlet form associated with the Ornstein-Uhlenbeck operator defined by (2.1). {Tt^{O-U}} and L denote the semigroup and the generator. L_{(p)} denotes the generator in L^{p} and we regard A_{(p)} as a perturbation of L_{(p)}.

Theorem 3.1. Under the condition (A.1), we have that Dom(A_{(2)}) = Dom(L_{(2)}) and further if D ⊆ Dom(L_{(2)}) is dense (with respect to the graph norm of L_{(2)}), the D is also dense in Dom(A_{(2)}). When p > 2, there exists a constant r = r_{p} such that if e^{|b|^{2}} ∈ L^{r+}, then the above result holds for A_{(p)}.

To prove this theorem, we recall a general theory. Let S be a closed operator on a Banach space B. We assume that S is hyper-dissipative (sometimes called m-dissipative) and hence generates a contraction semigroup. Here let us explain what it means that S is hyper-dissipative. An operator S on a Banach space B is called dissipative if for any u ∈ B and φ ∈ F (u) it holds that

⟨Su, φ⟩ ≤ 0.  (3.1)

Here F is the duality mapping, i.e., F (u) is a set of all φ such that

⟨u, φ⟩ = ∥u∥_{B}^{2} = ∥φ∥_{B^{∗}}^{2}.

When B = L^{p}, F is given by F (u) = |u|^{p−1} sgn(u)∥u∥_{p}^{2−p}. Further, if Ran(1 − S) = B, S is called hyper-dissipative.

Suppose we are given an operator T with Dom(T) ⊇ Dom(S). Then the following fact is well-known (e.g. Pazy [7, Theorem 3.3.2.]). If T satisfies the followings: (P.1) For any s ∈ [0, 1], (S + sT, Dom(S)) is dissipative. (P.2) There exists positive constants a and b such that 0 < a < 1 and

∥T x∥_{E} ≤ a∥Sx∥_{E} + b∥x∥_{E}, ∀x ∈ Dom(S).  (3.2)

Then (S + T, Dom(S)) is hyper-dissipative. Further if D ⊆ Dom(S) is dense, then D is dense in Dom(S + T) with respect to the graph norm.

To apply this fact, we first see that A_{(p)} + γ = L_{(p)} + b + γ is dissipative for some γ > 0. This fact was already shown when p = 2. In fact we can take ε = 0 in (2.6). When p > 2, let q be the conjugate exponent of p, i.e., 1/p + 1/q = 1. By the logarithmic Sobolev inequality and the Hausdorff-Young inequality st ≤ s log s − s + e^{t}, we have

∫_{B} |f|^{p}|b|^{2}dμ ≤ ∫_{B}\{\frac{2|f|^{p}}{q^{2}} log \frac{2|f|^{p}}{q^{2}} − \frac{2|f|^{p}}{q^{2}} + e^{q^{2}|b|^{2}/2}\}dμ

≤ \frac{2p}{q^{2}} ∫_{B}\{|f|^{p} log |f| + \frac{2}{q^{2}} log(2/q^{2})|f|^{p} − \frac{2}{q^{2}} |f|^{p} + e^{q^{2}|b|^{2}/2}\}dμ

≤ \frac{2p}{q^{2}} ∫_{B} |f|^{p} log(|f|/∥f∥_{p})dμ + \frac{2p}{q^{2}}∥f∥_{p}^{p} log ∥f∥_{p}

+ (\frac{2}{q^{2}} log(2/q^{2}) − \frac{2}{q^{2}} )∥f∥_{p}^{p} + ∥e^{q^{2}|b|^{2}/2}∥_{1}

≤ \frac{p^{2}}{q^{2}} ∫_{B} |Df|^{2}|f|^{p−2}dμ + \frac{2p}{q^{2}}∥f∥_{p}^{p} log ∥f∥_{p}

+ (\frac{2}{q^{2}} log(2/q^{2}) − \frac{2}{q^{2}} )∥f∥_{p}^{p} + ∥e^{q^{2}|b|^{2}/2}∥_{1}. (∵ (2.3))

Hence, if ∥f∥_{p} = 1, then, setting g = F (f) = |f|^{p−1} sgn(f)

−⟨Af, g⟩ = ⟨Df, Df⟩ − ⟨Df, gb⟩

= (p − 1) ∫_{B} |Df|^{2}|f|^{p−2}dμ − ⟨Df, b|f|^{p−1} sgn(f)⟩

≥ (p − 1) ∫_{B} |Df|^{2}|f|^{p−2}dμ − \frac{p}{2q} ∫_{B} |Df|^{2}|f|^{p−2}dμ − ∫_{B} \frac{q}{2p}|f|^{p}|b|^{2}dμ

= (p − 1 − \frac{p}{2q}) ∫_{B} |Df|^{2}|f|^{p−2}dμ − \frac{q}{2p}\{\frac{p^{2}}{q^{2}} ∫_{B} |Df|^{2}|f|^{p−2}dμ

+ \frac{2p}{q^{2}}∥f∥_{p}^{p} log ∥f∥_{p} + (\frac{2}{q^{2}} log(2/q^{2}) − \frac{2}{q^{2}} )∥f∥_{p}^{p} + ∥e^{q^{2}|b|^{2}/2}∥_{1}\}

= (p − 1 − \frac{p}{2q} − \frac{p}{2q}) ∫_{B} |Df|^{2}|f|^{p−2}dμ + \{(\frac{2}{q^{2}} log(2/q^{2}) − \frac{2}{q^{2}} ) + ∥e^{q^{2}|b|^{2}/2}∥_{1}\}

≥ \frac{2}{q^{2}} log(2/q^{2}) − \frac{2}{q^{2}} + ∥e^{q^{2}|b|^{2}/2}∥_{1}.

Therefore, for a general f ∈ L^{p} and g = F (f), it holds that

⟨Af, g⟩ + \{(\frac{2}{q^{2}} log(2/q^{2}) − \frac{2}{q^{2}} ) + ∥e^{q^{2}|b|^{2}/2}∥_{1}\}⟨f, g⟩ ≤ 0.  (3.3)

Now, it easily follows that A + γ is dissipative where we set

γ = \frac{2}{q^{2}} log(2/q^{2}) − \frac{2}{q^{2}} + ∥e^{q^{2}|b|^{2}/2}∥_{1}.  (3.4)

Next we see the condition (P.2). By the Hausdorff-young inequality,

|⟨b, Df⟩|^{2} ≤ |Df|^{2}|b|^{2} ≤ \frac{|Df|^{2}}{2(1 + ε)} log \frac{|Df|^{2}}{2(1 + ε)} − \frac{|Df|^{2}}{2(1 + ε)} + e^{2(1+ε)|b|^{2}}.

When ∥Df∥_{2} = 1, we have

∫_{B} |⟨b, Df⟩|^{2}dμ ≤ \frac{1}{2(1 + ε)} ∫_{B} |Df|^{2} log |Df|^{2}dμ − \frac{1}{2(1 + ε)} ∫_{B} |Df|^{2} log 2(1 + ε)dμ

− ∫_{B} \frac{|Df|^{2}}{2(1 + ε)}dμ + ∥e^{2(1+ε)|b|^{2}}∥_{1}

≤ \frac{1}{1 + ε} ∫_{B}(D^{2}f, D^{2}f)dμ − \frac{1}{2(1 + ε)}\{1 + log 2(1 + ε)\} + ∥e^{2(1+ε)|b|^{2}}∥_{1}

≤ \frac{1}{1 + ε} ∫_{B} |Lf|^{2}dμ − \frac{1}{2(1 + ε)}\{1 + log 2(1 + ε)\} + ∥e^{2(1+ε)|b|^{2}}∥_{1}.

Here we used the Logarithmic Sobolev inequality (2.2) for H^{∗}-valued function Df and the following inequality (see, e.g., [8, Proposition 4.5]):

∫_{E} |D^{2}f|^{2}dμ ≤ ∫_{E} |Lf|^{2}dμ.  (3.5)

We set

2γ = ∥e^{2(1+ε)|b|^{2}}∥_{1} − \frac{1}{2(1 + ε)} log 2e(1 + ε).  (3.6)

Then, for general f, we have

∫_{B} |⟨b, Df⟩|^{2}dμ ≤ \frac{1}{1 + ε} ∫_{B} |Lf|^{2}dμ + 2γ∥Df∥_{2}^{2}.

Now we note that there exists a constant K > 0 such that

2γ∥Df∥_{2}^{2} ≤ \frac{ε/2}{1 + ε}∥Lf∥_{2}^{2} + K∥f∥_{2}^{2}.

Therefore, we have

∫_{B} |⟨b, Df⟩|^{2}dμ ≤ \frac{1 + ε/2}{1 + ε}∥Lf∥_{2}^{2} + K∥f∥_{2}^{2}.

Thus we have

\{∫_{B} |⟨b, Df⟩|^{2}dμ\}^{1/2} ≤ \sqrt{\frac{1 + ε/2}{1 + ε}}∥Lf∥_{2} + \sqrt{K}∥f∥_{2}.

This completes the proof when p = 2.

When p > 2, we have to modify the above proof. We need the following proposition:

Proposition 3.2. There exist positive constants β, K_{1} and K_{2} such that

∫_{B} |Df|^{p} logp/2_{+} |Df|dμ ≤ β∥(λ − L)f∥_{p}^{p} + K_{1}∥Df∥_{p}^{p} logp/2_{+} ∥(λ − L)f∥_{p} + K_{2}∥Df∥_{p}^{p} (3.7)

where log_{+} x = max{log x, 0}.

To prove this theorem, we recall the Orlicz space. Let Φ be a strictly increasing convex function with Φ(0) = 0. We assume that Φ satisfies Δ_{2} condition. The Orlicz space L_{Φ} is the set of all function f satisfying

∫ Φ(|f|)dμ < ∞

with the norm

∥f∥_{Φ} := inf{λ > 0 ; ∫ Φ(|f|/λ)dμ ≤ 1}.  (3.8)

We take Φ to be equal to t^{p} log^{p/2} t for large t > 0. In this case, the space L_{Φ} is denoted by L log^{p/2} L. We also assume that Φ(t) ≥ t^{p} logp/2_{+} t for t > 0. On the Wiener space, it is known (see, Bakry-Meyer [1]) that \sqrt{λ − L}^{−1/2} is a bounded operator from L^{p}(μ) into L^{p} log^{p/2}(μ). This fact also holds for a vector valued function. Therefore we have that there exists a constant c such that

∥Df∥_{Φ} ≤ c∥\sqrt{λ − L}Df∥_{p}.

By combining this with the Meyer inequality, we get

∥Df∥_{Φ} ≤ κ∥(λ − L)f∥_{p}  (3.9)

for some constant κ. Now we are ready to prove Proposition 3.2.

Proof of Proposition 3.2. From (3.9), if κ∥(λ − L)f∥_{p} = 1,

∫_{B} |Df|^{p} logp/2_{+} |Df|dμ ≤ ∫_{B} Φ(|Df|)dμ ≤ 1.

For general f, taking f/κ∥(λ − L)f∥_{p} in the above argument, we have

∫_{B} \frac{|Df|^{p}}{κ^{p}∥(λ − L)f∥_{p}^{p}} logp/2_{+} \frac{|Df|}{κ∥(λ − L)f∥_{p}} dμ ≤ 1.

Now we use the inequality log_{+} ab ≤ log_{+} a + log_{+} b. By the Hölder inequality, for any ε > 0, there exists a constant K such that

(a + b)^{p/2} ≤ (1 + ε)a^{p/2} + Kb^{p/2}.  (3.10)

Together with this inequality, we have that for any ε > 0, there exists K such that

logp/2_{+} ab ≤ (1 + ε) logp/2_{+} a + K logp/2_{+} b.

Using this inequality, we get

logp/2_{+} |Df| = logp/2_{+}\{\frac{|Df|}{κ∥(λ − L)f∥_{p}} κ∥(λ − L)f∥_{p}\}

≤ (1 + ε) logp/2_{+} \frac{|Df|}{κ∥(λ − L)f∥_{p}} + K logp/2_{+} κ∥(λ − L)f∥_{p}.

Hence we have

∫_{B} \frac{|Df|^{p}}{κ^{p}∥(λ − L)f∥_{p}^{p}} \{\frac{1}{1 + ε} logp/2_{+} |Df| − \frac{K}{1 + ε} logp/2_{+} κ∥(λ − L)f∥_{p}\}dμ ≤ 1.

Thus we have

∫_{B} |Df|^{p} logp/2_{+} |Df|dμ ≤ (1 + ε)κ^{p}∥(λ − L)f∥_{p}^{p} + K∥Df∥_{p}^{p}{log_{+} ∥(λ − L)f∥_{p} + log_{+} κ}^{p/2}

which is the desired result.

Now we turn to the proof of Theorem 3.1 when p > 2. We take any α ≤ 2e. By the Hausdorff-Young inequality, we have

|⟨b, Df⟩|^{2} ≤ |Df|^{2}|b|^{2}

≤ \frac{α|Df|^{2}}{2} log \frac{α|Df|^{2}}{2} − \frac{α|Df|^{2}}{2} + e^{2|b|^{2}/α}

≤ α|Df|^{2} log |Df| + \{\frac{α}{2} log(α/2) − \frac{α}{2}\}|Df|^{2} + e^{2|b|^{2}/α}.

Then we have

|⟨b, Df⟩|^{2} ≤ α|Df|^{2} log_{+} |Df| + e^{2|b|^{2}/α}.

Using the inequality (3.10),

|⟨b, Df⟩|^{p} ≤ (1 + ε)α^{p/2}|Df|^{p} logp/2_{+} |Df| + Ke^{p|b|^{2}/α}.

Now by Proposition 3.2,

∫_{B} |⟨b, Df⟩|^{p}dμ ≤ (1 + ε)α^{p/2}β∥(λ − L)f∥_{p}^{p}

+ K_{3}∥Df∥_{p}^{p} logp/2_{+} ∥(λ − L)f∥_{p} + K_{4}∥Df∥_{p}^{p} + K∥e^{p|b|^{2}/α}∥_{1}.

Here β is the constant that appeared in (3.7). We can choose small δ > 0 so that

K_{3}δ^{p} logp/2_{+} t ≤ εα^{p/2}βt^{p}, t ≥ 0.

Therefore if ∥Df∥_{p} = δ, we have

∫_{B} |⟨b, Df⟩|^{p}dμ ≤ (1 + 2ε)α^{p/2}β∥(λ − L)f∥_{p}^{p} + K_{4}δ^{p} + K∥e^{p|b|^{2}/α}∥_{1}.

For general f, taking δf/∥Df∥_{p}, we have ∫_{B} |⟨b, δDf/∥Df∥_{p}⟩|^{p}dμ ≤ (1 + 2ε)α^{p/2}β∥(λ − L)δf/∥Df∥_{p}∥_{p}^{p} + K_{4}δ^{p} + K∥e^{p|b|^{2}/α}∥_{1}.

Thus we have,

∫_{B} |⟨b, Df⟩|^{p}dμ ≤ (1 + 2ε)α^{p/2}β∥(λ − L)f∥_{p}^{p} + ∥Df∥_{p}^{p}(K_{4}δ^{p} + K∥e^{p|b|^{2}/α}∥_{1})/δ^{p}.

If we take K_{5} sufficiently large, the following inequality holds

∥Df∥_{p}^{p}(K_{4}δ^{p} + K∥e^{p|b|^{2}/α}∥_{1})/δ^{p} ≤ εα^{p/2}β∥(λ − L)f∥_{p}^{p} + K_{5}∥f∥_{p}^{p}.

This inequality is known as the moment inequality. So eventually we have

∫_{B} |⟨b, Df⟩|^{p}dμ ≤ (1 + 3ε)α^{p/2}β∥(λ − L)f∥_{p}^{p} + K_{5}∥f∥_{p}^{p}.

If α^{p/2}β < 1, then we can see that (i) holds. To assure this, it is sufficient to assume that e^{|b|^{2}} ∈ L^{pβ^{2/p}+}. This exponent depends on the constant in Proposition 3.2.

4. Finite dimensional space

In this section, we consider a similar problem in the finite dimensional Euclidean space and we determine the domain of the generator. The generator that we consider is of the form A = Δ + b on the Euclidean space ℝ^{d}. Here, Δ is the Laplacian and b is a vector field, i.e., first order differential operator. The associated bilinear form E is given by

E(f, g) = ∫_{ℝ^{d}}{(∇f, ∇g) − (b, ∇f)g}dx.  (4.1)

We assume the following integrability condition for b: (A.2) There exists q ≥ d such that b ∈ L^{q}(ℝ^{d}). Under this condition, the form E defines a closed coercive semi-Dirichlet form (see Lyons-Zhang [3]).

In this section, we will determine the domain of A. As before, we denote the generator in L^{p} by A_{(p)} (p ≥ 2).

Theorem 4.1. Under the condition (A.2), we have that for any p < q, Dom(A_{(p)}) = Dom(Δ_{(p)}) and further if D ⊆ Dom(Δ_{(p)}) is dense (with respect to the graph norm of Δ_{(p)}), then D is also dense in Dom(A_{(p)}).

To prove this theorem, we need the following Sobolev inequality in ℝ^{d}. Let exponents p and r satisfies

\frac{1}{p} − \frac{1}{d} ≤ \frac{1}{r} ≤ \frac{1}{p}.

Then there exists a constant C such that

∥u∥_{r} ≤ C{∥∇u∥_{p}^{λ}∥u∥_{p}^{1−λ} + ∥u∥_{p}}  (4.2)

where λ = d(\frac{1}{p} − \frac{1}{r}). If \frac{1}{r} = \frac{1}{p} − \frac{1}{d}, then λ = 1 and (4.2) reads

∥u∥_{r} ≤ C{∥∇u∥_{p} + ∥u∥_{p}}.  (4.3)

If \frac{1}{r} > \frac{1}{p} − \frac{1}{d}, we have a stronger inequality: for any ε > 0 we can choose large K so that

∥u∥_{r} ≤ ε∥∇u∥_{p} + K∥u∥_{p}.  (4.4)

Even though this case is rather tractable, we use the Sobolev inequality of the form (4.3) to include the case \frac{1}{r} = \frac{1}{p} − \frac{1}{d}. We use the Sobolev inequality in this form.

Take any f ∈ H^{1,p}(ℝ^{d}), i.e., f, ∇f ∈ L^{p}. Set

h = |f|^{p/2} sgn(f).

Then ∇h = |f|^{(p/2)−1}∇f and

∥h∥_{2}^{2} = ∫_{ℝ^{d}} |f|^{p}dx,
∥∇h∥_{2}^{2} = ∫_{ℝ^{d}} |f|^{p−2}|∇f|^{2}dx.

Define r > 1 so that

\frac{1}{r} = \frac{1}{2} − \frac{1}{q}.  (4.5)

Since q ≥ d, it holds that \frac{1}{2} ≥ \frac{1}{r} ≥ \frac{1}{2} − \frac{1}{d}. Now, by the Sobolev inequality (4.3), we have

∥h∥_{r}^{2} ≤ C{∥∇h∥_{2}^{2} + ∥h∥_{2}^{2}}

which implies

{∫_{ℝ^{d}} |f|^{pr/2}dx}^{2/r}≤ C{∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx + ∫_{ℝ^{d}} |f|^{p}dx}.  (4.6)

Now we are ready to prove the dissipative property (P.1). For any f ∈ H^{2,p}(ℝ^{d}), g = |f|^{p−1} sgn(f), we have

−⟨Af, g⟩ = ∫_{ℝ^{d}}{(∇f, ∇g) − (∇f, b)|f|^{p−1} sgn(f)}dx

≥ (p − 1) ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx − ∫_{ℝ^{d}} |∇f||b||f|^{p−1}dx

≥ (p − 1) ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx − ∫_{ℝ^{d}} \frac{\sqrt{p − 1}}{2}|∇f||f|^{(p−2)/2} \frac{2}{\sqrt{p − 1}}|b||f|^{p/2}dx

≥ (p − 1) ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx

− 2\{\frac{p − 1}{4} ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx + \frac{4}{p − 1} ∫_{ℝ^{d}} |b|^{2}|f|^{p}dx\}

≥ \frac{p − 1}{2} ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx − \frac{8}{p − 1} ∫_{ℝ^{d}} |b|^{2}|f|^{p}dx

≥ \frac{p − 1}{2} ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx − \frac{8N^{2}}{p − 1} ∫_{ℝ^{d}} |f|^{p}dx − \frac{8}{p − 1} ∫_{ℝ^{d}} |b|^{2}1_{\{|b|>N\}}|f|^{p}dx.

Since \frac{1}{r/2} + \frac{1}{q/2} = 1, we can apply the Hölder inequality. Hence, by (4.6)

∫_{ℝ^{d}} |b|^{2}1_{{|b|>N}}|f|^{p}dx ≤ {∫_{ℝ^{d}} |b|^{2q/2}1_{{|b|>N}}dx}^{2/q}{∫_{ℝ^{d}} |f|^{pr/2}dx}^{2/r}

≤ C{∫_{ℝ^{d}} |b|^{q}1_{{|b|>N}}dx}^{2/q}{∫_{ℝ^{d}} |f|^{p−2}|∇f|^{2}dx + ∫_{ℝ^{d}} |f|^{p}dx}.

We take N to be large enough so that

\frac{8C}{p − 1}\{∫_{ℝ^{d}} |b|^{q}1_{\{|b|>N\}}dx\}^{2/q} ≤ \frac{p − 1}{4}.

Thus we have

−⟨Af, g⟩ ≥ \frac{p − 1}{4} ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx − \{\frac{8N^{2}}{p − 1} + \frac{p − 1}{4}\}∫_{ℝ^{d}} |f|^{p}dx.

Set γ = \frac{8N^{2}}{p−1} + \frac{p−1}{4}. Then we eventually obtain

−⟨Af, g⟩ + γ⟨f, g⟩ ≥ \frac{p − 1}{4} ∫_{ℝ^{d}} |∇f|^{2}|f|^{p−2}dx ≥ 0

which implies that A − γ is dissipative.

Next we will see (P.2). We set \frac{1}{α} = \frac{p}{q} and \frac{1}{β} = 1 − \frac{p}{q}. Since p < q, we have 1 < α, β. Note that

\frac{1}{pβ} = \frac{1}{p} − \frac{1}{q} ≥ \frac{1}{p} − \frac{1}{d}.

Hence, by the Sobolev inequality, we have

{∫_{ℝ^{d}} |∇f|^{pβ}dx}^{1/β} ≤ C{∥∇^{2}f∥_{p}^{p} + ∥∇f∥_{p}^{p}}
≤ C′{∥Δf∥_{p}^{p} + ∥∇f∥_{p}^{p}}.

Now, by the Hölder inequality, we have

∫_{ℝ^{d}} |(b, ∇f)|^{p}dx ≤ ∫_{ℝ^{d}} |b|^{p}1_{{|b|>N}}|∇f|^{p}dx + N^{p} ∫_{ℝ^{d}} |∇f|^{p}dx

≤ {∫_{ℝ^{d}} |b|^{pα}1_{{|b|>N}}dx}^{1/α}{∫_{ℝ^{d}} |∇f|^{pβ}dx}^{1/β}+N^{p} ∫_{ℝ^{d}} |∇f|^{p}dx

≤ C′{∫_{ℝ^{d}} |b|^{q}1_{{|b|>N}}dx}^{1/α}{∥Δf∥_{p}^{p} + ∥∇f∥_{p}^{p}} + N^{p}∥∇f∥_{p}^{p}.

For any ε > 0, we take N to be large enough so that

C′{∫_{ℝ^{d}} |b|^{q}1_{{|b|>N}}dx}^{1/α}≤ ε

Moreover, by the interpolation inequality, there exist a constant K such that

(ε + N^{p})∥∇f∥_{p}^{p} ≤ ε∥Δf∥_{p}^{p} + K∥f∥_{p}^{p}.

Thus we have

∫_{ℝ^{d}} |(b, ∇f)|^{p}dx ≤ 2ε∥Δf∥_{p}^{p} + K∥f∥_{p}^{p}

which implies (P.2). This completes the proof.

References

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[2] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061–1083.

[3] T. Lyons and T. Zhang, Convergence of non-symmetric Dirichlet processes, preprint,

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[6] Z.-M. Ma and M. Röckner, Markov processes associated with positivity preserving coercive forms, Canadian J. Math., 47 (1995), 817–840.

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Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan

  E-mail address: ichiro@kusm.kyoto-u.ac.jp