Log-Harnack inequality for stochastic Burgers equations and applications

J. Math. Anal. Appl. 384 (2011) 151–159

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Log-Harnack inequality for stochastic Burgers equations and applications ✩ Feng-Yu Wang a,b , Jiang-Lun Wu b , Lihu Xu c,∗ a b c

School of Math. Sci. and Lab. Math. Com. Sys., Beijing Normal University, Beijing 100875, China Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands

a r t i c l e

i n f o

a b s t r a c t By proving an L 2 -gradient estimate for the corresponding Galerkin approximations, the logHarnack inequality is established for the semigroup associated to a class of stochastic Burgers equations. As applications, we derive the strong Feller property of the semigroup, the irreducibility of the solution, the entropy-cost inequality for the adjoint semigroup, and entropy upper bounds of the transition density. © 2011 Elsevier Inc. All rights reserved.

Article history: Received 20 October 2010 Available online 16 February 2011 Submitted by Goong Chen Keywords: Stochastic Burgers equation Log-Harnack inequality Strong Feller property Irreducibility Entropy-cost inequality

1. Introduction Let T = R/(2π Z) be equipped with the usual Riemannian metric, and let dθ denote the Lebesgue measure on T. Then

   H := x ∈ L 2 (dθ): x(θ) dθ = 0 T

is a separable real Hilbert space with inner product and norm

 x, y  :=

x(θ) y (θ) dθ,

x := x, x1/2 .

T

For x ∈ C (T), the Laplacian operator  is given by x = x . Let ( A , D( A )) be the closure of (−, C 2 (T) ∩ H) in H, which is a positively definite self-adjoint operator on H. Then 2





V := D A 1/2 ,

  x, y  V := A 1/2 x, A 1/2 y

gives rise to a Hilbert space, which is densely and compactly embedded in H. By the integration by parts formula, for any ✩ The first author is supported in part by WIMCS and NNSFC (10721091). The third author would like to gratefully thank EURANDOM and Hausdorff Research Institute for Mathematics for providing nice research environment. His work is partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 258237. We would like to thank Arnaud Debussche for the stimulating discussions. We also would like to thank the anonymous referee for carefully reading the paper and many nice advices for improving the paper. Corresponding author.

*

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.02.032

©

2011 Elsevier Inc. All rights reserved.

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x ∈ C 2 (T) we have

2 x2V := A 1/2 x =



 (x Ax)(θ) dθ =

T

 2

x (θ) dθ.

T

Moreover, for x, y ∈ C (T) ∩ H, set B (x, y ) := xy  . Then B extends to a unique bounded bilinear operator B : V × V → H with (see Proposition 2.1 below) 1

 B  V →H :=

sup

x V , y  V 1

√ B (x, y )  π .

(1.1)

Consider the following stochastic Burgers equation



dX t = −



ν A Xt + B ( Xt ) dt + Q dW t ,

(1.2)

where ν > 0 is a constant, B (x) := B (x, x) for x ∈ V , Q is a Hilbert–Schmidt operator on H, and W t is the cylindrical Brownian motion on H. According to [3, Chapter 5] (see also [5, Theorem 14.2.4]), for any x ∈ H, this equation has a unique solution with the initial condition X 0 = x, which is a continuous Markov process on H and is denoted by X tx from now on. If moreover x ∈ V , then X tx is a continuous process on V (see Proposition 2.3 below). We are concerned with the associated Markov semigroup P t given by





P t f (x) := E f X tx ,

x ∈ H, t  0

for f ∈ Bb (H), the set of all bounded measurable functions on H. The purpose of this paper is to investigate regularity properties of P t , such as strong Feller property, heat kernel upper bounds, contractivity properties, and entropy-cost inequalities. To do this, a powerful tool is the dimension-free Harnack inequality introduced in [12] for diffusions on Riemannian manifolds (see also [1,2] for further development). In recent years, this inequality has been established and applied intensively in the study of SPDEs (see e.g. [9,13,8,6,7,15] and references within). In general, this type of Harnack inequality can be formulated as

  

( P t f )α (x)  P t f α ( y ) exp C α (t , x, y ) ,

f  0,

(1.3)

where α > 1 is a constant, C α is a positive function on (0, ∞) × H2 with C α (t , x, x) = 0, which is determined by the underlying stochastic equation. On the other hand, in some cases this kind of Harnack inequality is not available, so that the following weaker version (i.e. the log-Harnack inequality)

P t log f (x)  log P t f ( y ) + C (t , x, y ),

f 1

(1.4)

becomes an alternative tool in the study. In general, according to [14, Section 2], (1.4) is the limit version of (1.3) as α → ∞. This inequality has been established in [10,14], respectively, for semi-linear SPDEs with multiplicative noise and the Neumann semigroup on non-convex manifolds. As for the stochastic Burgers equation (1.2), by using A 1+σ for σ > 12 to replace A (i.e. the hyperdissipative equation is concerned), the first and the third named authors established an explicit Harnack inequality of type (1.3) in [16], where a more general framework, which includes also the stochastic hyperdissipative Navier–Stokes equations, was considered. But, when σ  12 , the known arguments (i.e. the coupling argument and gradient estimate) to prove (1.3) are no longer valid. Therefore, in this paper we turn to investigate the log-Harnack inequality for P t associated to (1.2), which also provides some important regularity properties of the semigroup (see Corollary 1.2 below). Note that the stochastic Burgers equation does not satisfy the Lipschitz and monotone conditions required in [10], the present study cannot be covered there. To state our main result, we introduce the intrinsic norm induced by the diffusion part of the solution. For any x ∈ H , let

 x Q := inf zH : z ∈ H, Q ∗ z = x , where Q ∗ is the adjoint operator of Q , and we take x Q = ∞ if the set in the right-hand side is empty. Moreover, let  ·  and  · HS denote the operator norm and the Hilbert–Schmidt norm respectively for bounded linear operators on H. Theorem 1.1. Assume that ν 3  4π  A −1/2 Q 2 . Then for any f ∈ Bb (H) with f  1,

P t log f (x)  log P t f ( y ) + holds for t > 0 and x, y ∈ H.

2π  Q 2HS x − y 2Q

ν 2 [1 − exp(− 4νπ2  Q 2HS t )]

 exp

4π 

ν2

x2 ∨  y 2



 (1.5)

F.-Y. Wang et al. / J. Math. Anal. Appl. 384 (2011) 151–159

153

Before introducing consequences of Theorem 1.1, let us recall that the invariant probability measure of P t exists, and any invariant probability measure μ satisfies μ( V ) = 1. These follow immediately since V is compactly embedded in H and due to the Itô formula one has

2 E X t0 H + 2ν

t

2 E X s0 V ds   Q 2HS t ,

t  0.

0

Next, for any two probability measures function

μ1 , μ2 on H, let W c (μ1 , μ2 ) be the transportation-cost between them with cost 

(x, y ) → c(x, y ) := x − y 2Q exp

4π 

ν2

x2 ∨  y 2

 

.

That is,

 W c (μ1 , μ2 ) =

c(x, y ) μ(dx, d y ),

inf

μ∈C (μ1 ,μ2 )

H×H

where C (μ1 , μ2 ) is the set of all couplings of

B V (x, r ) = z ∈ V :  z − x V < r , 

μ1 and μ2 . Finally, let

x ∈ V , r > 0.

Corollary 1.2. Assume that ν 3  4π  A −1/2 Q 2 . (1) For any t > 0, P t is intrinsic strong Feller, i.e.

lim

x− y  Q →0

P t f ( y ) = P t f (x),

x ∈ H, f ∈ Bb (H).

(2) Let μ be an invariant probability measure of P t and let P t∗ be the adjoint operator of P t w.r.t. μ. Then the entropy-cost inequality







μ P t∗ f log P t∗ f 

1

2π  Q 2HS

ν 1 − exp[− 4π2  Q 2HS t ] ν 2

W c ( f μ, μ),

f  0,

μ( f ) = 1

holds for all t > 0. (3) Let  ·  Q  C  ·  V hold for some constant C > 0. Then

 y  P X t ∈ B V (x, r ) > 0,

x, y ∈ V , t , r > 0.

(1.6)

Consequently, P t has a unique invariant probability measure μ, which is fully supported on V , i.e. any non-empty open set G ⊂ V . Furthermore, μ is strong mixing, i.e. for any f ∈ Bb (H),

lim P t f (x) = μ( f ),

t →∞

μ( V ) = 1 and μ(G ) > 0 for

∀x ∈ V .

(4) Under the same assumption as in (3), P t has a transition density pt (x, y ) w.r.t. μ on V such that the entropy inequalities

 pt (x, z) log V

pt (x, z) pt ( y , z)

μ(dz) 

1

2π  Q 2HS c(x, y )

(1.7)

ν 2 1 − exp[− 4π2  Q 2HS t ] ν

and



 pt (x, y ) log pt (x, y ) μ(d y )  − log

V

 exp −

V

1

2π  Q 2HS c(x, y )

ν 2 1 − exp[− 4π2  Q 2HS t ]



μ(d y )

(1.8)

ν

hold for all t > 0 and x, y ∈ V . To prove the above results, we present in Section 2 some preparations including a brief proof of (1.1), a convergence theorem for the Galerkin approximation of (1.2), and the continuity of the solution in V . Finally, complete proofs of Theorem 1.1 and Corollary 1.2 are addressed in Section 3.

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F.-Y. Wang et al. / J. Math. Anal. Appl. 384 (2011) 151–159

2. Some preparations Obviously, (1.1) is equivalent to the following result. Proposition 2.1.  B (x, y )2  π x2V  y 2V holds for any x, y ∈ V .



Proof. We shall take the continuous version for an element in V . Since T x(θ) dθ = 0, there exists θ0 ∈ T such that x(θ0 ) = 0. For any θ ∈ T, let γ : [0, d(θ0 , θ)] → T be the minimal geodesic from θ0 to θ , where d(θ0 , θ)( π ) is the Riemannian distance between these two points. By the Schwartz inequality we have

d(θ

2 

 0 ,θ ) d





2

x(θ) 2 =

x(γs ) ds  d(θ0 , θ) x (ξ ) dξ  π x2V .



ds T

0

Therefore,

B (x, y ) 2 =



   2

xy (θ) dθ  π x2  y 2 . V V

2

T

Remark. From the proof we see that (1.1) is a property in one-dimension, since for d  2 there is no any constant C ∈ (0, ∞) such that



x2∞  C



|∇ x|2 (θ) dθ,

x ∈ C 1 Td



Td

holds. The invalidity of (1.1) in high dimensions is the main reason why we only consider here the stochastic Burgers equation rather than the stochastic Navier–Stokes equation. Next, due to the fact that to prove the log-Harnack inequality we have to apply the Itô formula for a reasonable class of reference functions which is, however, not available in infinite-dimensions, we need to make use of the finite-dimensional approximations. To introduce the Galerkin approximation of (1.2), let us formulate H by using the standard ONB {ek : k ∈ Z} for the complex Hilbert space L 2 (T → C; dθ), where

1 ek (θ) := √ eikθ , 2π

θ ∈ T.

Obviously, ek = −k2 ek holds for all k ∈ Z, and an element



x :=

xk ek ,

xk ∈ C

k∈Z

ˆ := Z \ {0}, and belongs to H if and only if x0 = 0, x¯ k = x−k for all k ∈ Z



H=

xk ek : x¯ k = x−k ,

ˆ k∈Z



 |xk |2 < ∞ .



ˆ k ∈Z

|xk |2 < ∞. Therefore,

ˆ k∈Z

For any m ∈ N, let

 Hm = x ∈ H: x, ek  = 0 for |k| > m , which is a finite-dimensional Euclidean space. Let πm : H → Hm be the orthogonal projection. Let B m = Q m = πm Q . Consider the following stochastic differential equation on Hm : (m)

dX t

 (m)   (m) = − ν A Xt + B m Xt dt + Q m dW t .

πm B and (2.1)

Since coefficients in this equation are smooth and



(m) 2

d X t

 (m)   2 Q m 2HS dt + 2 X t , Q m dW t , m,x

we conclude that starting from any x ∈ Hm this equation has a unique strong solution X t (m)

Pt



m,x 

f (x) = E f X t

,

t  0, x ∈ Hm , f ∈ Bb (Hm ).

In the spirit of [3, Theorem 5.7], the next result implies

which is non-explosive. Let

F.-Y. Wang et al. / J. Math. Anal. Appl. 384 (2011) 151–159

(m)

P t f (x) = lim P t m→∞

f (xm ),

155

x ∈ H, f ∈ C b (H)

(2.2)

for {xm ∈ Hm }m1 such that xm → x in H. m,xm

Proposition 2.2. For any {xm ∈ Hm }m1 such that x − xm  H → 0, we have  X tx − X t quently, (2.2) holds. m,xm

Proof. Simply denote X t (m) = X t

t E



 → 0 in probability as m → ∞. Conse-

and X t = X tx . It is easy to see that

2   X s 2V + X s (m) V ds  C (1 + t )

(2.3)

0

holds for some constant C > 0. By the Itô formula we have

X t − X t (m) 2  −2

t

2      ν X s − X s (m) V + B ( X s ) − B X s (m) , X s − X s (m) ds + ηt (m),

0

where

ηt (m) :=  Q −

Q m 2HS t

r

 



+ x − xm  + 2 sup X s − X s (m), ( Q − Q m ) dW s ,

r ∈[0,t ]

2

0

which goes to 0 a.s. as m → ∞. Since by (1.1)

   

B (x) − B ( y ), x − y = B (x, x − y ) + B (x − y , y ), x − y

   π x − y  x V +  y  V x − y  V , it follows from (2.4) that

X r − X r (m) 2  π

r

ν

  X s − X s (m) 2  X s 2 + X s (m) 2 ds + ηt (m), V V

r ∈ [0, t ].

0

Therefore,

 t  2   X t − X t (m) 2  ηt (m) exp π  X s 2V + X s (m) V ds .

ν

0

Combining this with (2.3) we obtain that for any N > 0,

 t  2 2     C (1 + t ) N π /ν 2 P  X s  V + X s (m) V ds  N  P X t − X t (m)  ηt (m)e N

0

which goes to 0 as N → ∞. Observe that

2 2     P X t − X t (m)  ηt (m)e N π /ν  P X t − X t (m)  e− N π /ν , ηt (m)  e−2N π /ν . The previous two inequalities imply

2  C (1 + t )    P X t − X t (m)  e− N π /ν  + P ηt (m)  e−2N π /ν . N

Since

ηt (m) → 0 as m → ∞, this implies that  X t − Xt (m) → 0 in probability as m → ∞. 2

Finally, we have the following result for the continuity of the solution in V . Proposition 2.3. For any x ∈ V , X tx is a continuous process in V .

(2.4)

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F.-Y. Wang et al. / J. Math. Anal. Appl. 384 (2011) 151–159

Proof. For fixed x ∈ V and T > 0, we introduce the map









Y : C [0, T ]; V → C [0, T ]; V , such that for any u ∈ C ([0, T ]; V ), {Y t (u )}t ∈[0, T ] solves the deterministic equation



Y˙ t (u ) = −





ν AY t (u ) + B Y t (u ) + ut ,

Y 0 (u ) = x.

(2.5)

Then Y (u ) ∈ C ([0, T ]; V ), see e.g. [11, Theorem 3.2] (the theorem is for 2D Navier–Stokes equation, and the proof works also for our case). Next, let

t Zt =

e−ν (t −s) A Q dW s .

0

Since Q is Hilbert–Schmidt on H, Z t is a continuous process on V (see e.g. [4, Theorem 5.9]). Therefore, X tx = Y t ( Z ) + Z t is also continuous in V . 2 3. Proofs of Theorem 1.1 and Corollary 1.2 (m)

According to [10], the key step to prove the log-Harnack inequality for P t



 

Q m D P (m) f 2 (x)  P (m) | Q m D f |2 (x)C (t , x), t

is the L 2 -gradient estimate

f ∈ C b1 (Hm )

t

for some continuous function C on (0, ∞) × Hm , where D is the gradient operator on Hm , i.e. for any f ∈ C 1 (Hm ), the element D f (x) ∈ Hm is determined by





D f (x), h = D h f (x) := lim

f (x + εh) − f (x)

ε →0

ε

h ∈ Hm .

,

To derive the desired gradient estimate, we need the following lemma. Lemma 3.1. For any x ∈ Hm and t  0,

 E exp



ν

m,x 2 X + ν

2 A −1/2 Q m 2

t

t

m , x 2 X ds s V



  exp

0

ν (x2 +  Q m 2HS t ) 2 A −1/2 Q m 2

 .

Proof. By the Itô formula and easy fact x, B m (x) = 0 [11], we have





d X t

m , x 2

2   + 2ν X tm,x V dt =  Q m 2HS dt + 2 X tm,x , Q m dW t .

(3.1)

Let









τn = inf t  0: Xtm,x  n , n ∈ N. τn → ∞ as n → ∞. Let

We have

(n)

Mt

t∧τn



=



X sm,x , Q m dW s .

0

Then for any λ > 0

(n)

t → exp 2λ M t

  − 2λ2 M (n) t

is a martingale. Therefore, it follows from (3.1) that t∧τn t∧τn   2 m,x 2 ∗ m , x 2 X ds − 2λ2 Q X ds E exp λ X tm∧,τxn + 2ν λ s m s V 0

0

    (n)  E exp λ x2 + t  Q m 2HS + 2λ M t − 2λ2 M (n) t

  = exp λ x2 + t  Q m 2HS .

(3.2)

F.-Y. Wang et al. / J. Math. Anal. Appl. 384 (2011) 151–159

157

Noting that

∗ ∗ −1/2 1/2 ∗ −1/2 Q x = Q A · x V = A −1/2 Q m · x V , A x  Q m A m m

x ∈ Hm ,

by letting n ↑ ∞ in (3.2) and taking

ν

λ=

,

2 A −1/2 Q m 2

2

we complete the proof.

Lemma 3.2. Let ν 3  4π  A −1/2 Q m 2 . Then for any f ∈ C b1 (Hm ),

      Q m D P (m) f 2 (x)  P (m)  Q m D f 2 (x) exp 2π x2 + t  Q m 2 t

t

HS

ν2

holds for all t  0 and x ∈ Hm . Proof. Let h ∈ Hm . According to e.g. [3, Section 5.4], m,x

m,x+εh

Xt

:= lim

D h Xt

− X tm,x

ε

ε →0

,

t0

exists and solves the ordinary differential equation

d dt

m,x

D h Xt

   = − ν A D h X tm,x + B˜ m X tm,x , D h X tm,x ,

where B˜ m (x, y ) := B (x, y ) + B ( y , x) for x, y ∈ Hm . By (1.1), this implies that

d dt

   D h X m,x 2 = −2ν A D h X m,x 2 − 2 A D h X m,x , B˜ m X m,x , D h X m,x t t t t t V 

1 2ν

  B˜ m X m,x , D h X m,x 2  2π X m,x 2 D h X m,x 2 . t t t t V V

ν

Therefore,

  t 2 2 π 2 m , x D h X  h2 exp X m,x ds . s t V V V

ν

0

Since

ν

3

 4π  A −1/2 Q

ν2 2 A −1/2 Q m 2

m



2

implies that

2π

ν

,

by Lemma 3.1 and using the Jensen inequality we arrive at

  2  2π  x2 + t  Q m 2HS . E D h X tm,x V  h2V exp

(3.3)

ν

Thus,

 E

m,x+εh

f ( Xt

) − f ( X tm,x )

ε

 ε

2 =E

1

ε



Df



 m,x+sh  , D h X tm,x+sh ds Xt

2

0



h2V  D f 2∞



ε exp

ε 0

 Therefore,

m,x+εh

f ( Xt

)− f ( X tm,x )

ε

h2V  D f 2∞ exp



2π 

ν

4π 

ν

x + sh2 + t  Q m 2HS 2

2

x + h

+ t  Q m 2HS

 



ds

 ,

ε ∈ (0, 1].

is uniformly integrable, this, combining with the dominated convergence theorem, implies

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F.-Y. Wang et al. / J. Math. Anal. Appl. 384 (2011) 151–159

 (m)

Dh Pt

f (x) = lim E ε →0



= E Df



m,x+εh

f ( Xt

) − f ( X tm,x )



ε

 m,x  , D h X tm,x , Xt

f ∈ C b1 (Hm ), x ∈ Hm , t  0.

(3.4)

On the other hand, we have

    Q m D P (m) f 2 = sup Q m D P (m) f , h˜ 2 = sup D P (m) f , Q ∗ h˜ 2 m t t t h˜ 1

= where

(m) 2 sup D h P t f ,

h˜ 1

(3.5)

h Q m 1

 h Q m := inf z: z ∈ Hm , Q m∗ z = h

and h Q m = ∞ if the set on the right-hand side is empty. Now, for any h ∈ Hm with h Q m  1, let { zn }n1 ⊂ H be such ∗ z = h and  z   1 + 1 . By (3.4) we have that Q m n n n



         

D h P (m) f 2 (x) = E D f X m,x , D h X m,x 2 = E Q m D f X m,x , D z X m,x 2 n t t t t t 2    2    2    E Q m D f X m , x E D z X m , x = E Q m D f X m , x E D t

n

t

t

Combining this with (3.3) and (3.5) and letting n ↑ ∞, we complete the proof.

m , x 2 . A −1/2 zn X t V

2 (m)

According to the L 2 -gradient estimate in Lemma 3.2, we are able to prove the log-Harnack inequality for P t

as in [10].

Proposition 3.3. Let ν 3  4π  A −1/2 Q m 2 . For any f ∈ Bb (Hm ) with f  1, 2π  Q m 2HS x − y 2Q m exp[ 4νπ2 (x2 ∨  y 2 )] (m) (m) P t log f (x)  log P t f ( y ) + ν 2 [1 − exp(− 4νπ2  Q m 2HS t )] holds for all t > 0 and x, y ∈ Hm . ∗ z = x − y and  z 2  x − y 2 + 1 . Let Proof. It suffices to prove for x − y  Q m < ∞. Let { zn } ⊂ Hm be such that Q m n n Qm n

γ ∈ C 1 ([0, t ]; R) such that γ (0) = 0, γ (t ) = 1. Finally, let xs = (x − y )γ (s) + y, s ∈ [0, t ]. Then, by Lemma 3.2 we have (see [10, Proof of Theorem 2.1] for explanation of the second equality) (m)

Pt

(m)

log f (x) − log P t

t =

f ( y)

d  (m) (m) P s log P t −s f (xs ) ds

ds 0

t  =



 1 m) 2 m)  (xs ) ds + γ  (s) x − y , D P s(m) log P t(− − P s(m) Q m D log P t(− sf sf 2

0



t 

(m)

Ps





2 (m) (m) 2π (xs 2 + Q m 2HS s)/ν 2



Q m D log P t −s f (xs ) ds − Q m D log P t −s f + γ (s) · zn e 1 2

0



zn 2

t

2

 2 4π (x 2 + Q 2 s)/ν 2 s m HS

γ (s) e ds.

0

Since xs   x ∨  y , by taking 1 − exp[− 4νπ2  Q m 2HS s] γ (s) = , 1 − exp[− 4νπ2  Q m 2HS t ]

s ∈ [0, t ]

we obtain (m)

Pt

(m)

log f (x) − log P t

f ( y) 

1

2π  Q 2HS  zn 2

ν 2 1 − exp[− 4π2  Q 2HS t ] ν

This completes the proof by letting n → ∞.

2

 exp

4π 

ν2

x2 ∨  y 2



 .

F.-Y. Wang et al. / J. Math. Anal. Appl. 384 (2011) 151–159

159

Proof of Theorem 1.1. It suffices to prove for f ∈ C b (H) with f  1. Let x − y  Q < ∞. For any ε > 0, let z ∈ H such ∗ z = π x − π y. Let x = π x, z = π z and that Q ∗ z = x − y and  z2  x − y 2Q + ε . For any m ∈ N, we have Q m m m m m m m ∗ ( z − π z). Then z ∈ H and Q ∗ z = x − y , so that ym = πm y + Q m m m m m m m m

xm − ym 2Q m  zm 2  x − y 2Q + ε . Moreover, it is easy to see that xm → x and ym → y hold in H. Combining these with Proposition 3.3 and (2.2), and letting m → ∞ and ε → 0, we complete the proof. 2 Proof of Corollary 1.2. The intrinsic strong Feller property follows from [14, Proposition 2.3], while the entropy-cost inequality in (2) follows from the proof of Corollary 1.2 in [10]. So, it remains to prove (3) and (4). (a) Applying (1.5) to f := 1 + m1 B (x,r ) for m  1, we obtain





P t log(1 + m1 B V (x,r ) )(x)  log 1 + m P t 1 B V (x,r ) ( y ) + α (t )c(x, y ),

t > 0, m  1

for some function α : (0, ∞) → (0, ∞) independent of x, y and m. By Proposition 2.3 we have Then there exists t 0 > 0 depending only on x such that

 1  P X tx − x V < r  , 2

(3.6)

 X tx

− x V → 0 as t → 0.

t ∈ [0, t 0 ].

y

Thus, if P( X t ∈ B V (x, r )) = 0 for some t ∈ (0, t 0 ], then (3.6) yields that

1 2

log(1 + m)  P t log(1 + m1 B V (x,r ) )(x)  α (t )c(x, y ),

m  1,

which is impossible since  ·  Q  C x − y  V implies that c(x, y ) < ∞ for x, y ∈ V . Therefore,

  P X tz ∈ B V (x, r ) > 0,

t ∈ (0, t 0 ], z ∈ V .

Combining this with the Markov property we see that for t > t 0 ,

 y  P X t ∈ B (x, r ) =



  P X tz0 ∈ B (x, r ) P t −t0 ( y , dz) > 0,

V y

where P t −t0 ( y , dz) is the distribution of X t −t0 . Therefore, (1.6) holds. (b) Since (1) and  ·  Q  C  ·  V imply the strong Feller property of P t on V , by the Doob Theorem, see e.g. [5, Theorem 4.2.1], P t has a unique invariant measure μ on V . The full support property of μ, together with the strong Feller of P t , implies the existence of transition density pt (x, y ). Finally, due to [14, Proposition 2.4(2)], (1.7) is equivalent to the log-Harnack inequality (1.5), while (1.8) follows from (1.5) according to the proof of [10, Corollary 1.2]. 2 References [1] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. Sci. Math. 130 (2006) 223–233. [2] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds, Stochastic Process. Appl. 119 (2009) 3653–3670. [3] G. Da Prato, Kolmogorov Equations for Stochastic PDEs, Adv. Courses Math. CRM Barcelona, Birkhäuser Verlag, Basel, 2004. [4] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. [5] G. Da Prato, J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Math. Soc. Lecture Note Ser., vol. 229, Cambridge University Press, Cambridge, 1996. [6] G. Da Prato, M. Röckner, F.-Y. Wang, Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups, J. Funct. Anal. 257 (2009) 992–1017. [7] A. Es-Sarhir, M.-K.v. Renesse, M. Scheutzow, Harnack inequality for functional SDEs with bounded memory, Electron. Comm. Probab. 14 (2009) 560– 565. [8] W. Liu, F.-Y. Wang, Harnack inequality and strong Feller property for stochastic fast-diffusion equations, J. Math. Anal. Appl. 342 (2008) 651–662. [9] M. Röckner, F.-Y. Wang, Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal. 203 (2003) 237–261. [10] M. Röckner, F.-Y. Wang, Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010) 27–37. [11] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995, xiv+141 pp. [12] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417–424. [13] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333–1350. [14] F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl. 94 (2010) 304–321. [15] F.-Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on non-convex manifolds, Ann. Probab., doi:10.1214/10-aop600, in press, arXiv:0911.1644. [16] F.-Y. Wang, L. Xu, Bismut type formula and its application to stochastic hyperdissipative Navier–Stokes/Burgers equations, arXiv:1009.1464.