aa r X i v : . [ m a t h . P R ] J un Integrability Conditions for SDEs andSemi-Linear SPDEs ∗ Feng-Yu Wang
Laboratory of Mathematical and Complex Systems, Beijing Normal University, Beijing 100875, ChinaDepartment of Mathematics, Swansea University, Singleton Park, SA2 8PP, UKEmail: [email protected] ; [email protected]
September 28, 2018
Abstract
By using the local dimension-free Harnack inequality established on incompleteRiemannian manifolds, integrability conditions on the coefficients are presented forSDEs to imply the non-explosion of solutions as well as the existence, uniqueness andregularity estimates of invariant probability measures. These conditions include a classof drifts unbounded on compact domains such that the usual Lyapunov conditions cannot be verified. The main results are extended to second order differential operatorson Hilbert spaces and semi-linear SPDEs.
AMS subject Classification: 60H15, 60J45.Keywords: Non-explosion, invariant probability measure, local Harnack inequality, SDE,SPDE.
In recent years, the existence and uniqueness of strong solutions up to life time have beenproved under local integrability conditions for non-degenerate SDEs, see [6, 25, 22, 44, 45]and references within. See also [14, 15, 16, 39, 41, 42] for extensions to degenerate SDEs andsemi-linear SPDEs.As a further development in this direction, the present paper provides reasonable integra-bility conditions for the non-explosion of solutions, as well as the existence, uniqueness andregularity estimates of invariant probability measures. An essentially new point in the study ∗ Supported in part by NNSFC(11131003, 11431014), the 985 project.
1s to make use of a local Harnack inequality in the spirit of [32]. With this inequality we areable to prove the non-explosion of a weak solution constructed from the Girsanov transform,see the proof of Lemma 3.1 below for details. Moreover, we use the hypercontractivity of thereference Markov semigroup to prove the boundedness of a Feyman-Kac semigroup inducedby the singular SDE under study, which enables us to prove the existence of the invariantprobability measure as well as a formula for the derivative of the density, see (4.3) and theproof of Lemma 4.2 below for details. To explain the motivation of the study more clearly,below we first recall some existing results in the literature, then present a simple exampleto show how far can we go beyond.Let W t be the d -dimensional Brownian motion on a complete filtration probability space(Ω , F , { F t } t ≥ , P ). Consider the following SDE (stochastic differential equation) on R d :(1.1) d X t = b ( X t )d t + √ σ ( X t )d W t , where b : R d → R d is measurable and σ ∈ W , loc ( R d → R d ⊗ R d ; d x ) such that σ ( x ) is invertiblefor every x ∈ R d . According to [45, Theorem 1.1] (see also [6, 25, 22, 44] for earlier results),if | b | + k∇ σ k ∈ L ploc (d x ) for some p > d , then for any initial point x the SDE (1.1) has aunique solution ( X xt ) t ∈ [0 ,ζ x ) up to life time ζ x . We note that in [45] the global integrabilityand the uniform ellipticity conditions are assumed, but these conditions can be localizedsince for the existence and uniqueness up to life time one only needs to consider solutionsbefore exiting bounded domains. On the other hand, the ODEd X t = b ( X t )d t does not have pathwise uniqueness if b is merely H¨older continuous (for instance, d = 1 and b ( x ) := | x | α for some α ∈ (0 , σ is bounded and(1.2) | b | ≤ C + F for some constant C > F ∈ L p (d x ) for some p > d, then the solution to (1.1) is non-explosive. As the Lebesgue measure is infinite, this conditionis very restrictive. So, one of our aims is to replace it by integrability conditions with respectto a probability measure, see Theorem 2.1 and Corollary 2.2 below.We would like to indicate that when the invariant measure µ is given, there exist criteriaon the conservativeness of non-symmetric Dirichlet forms, which imply the non-explosion ofsolutions for µ -a.e. initial points, see [31] and references within. However, in our study theinvariant probability measure is unknown, which is indeed the main object to characterize.In general, to prove the existence of invariant probability measures one uses Lyapunov (ordrift) conditions. For instance, if there exists a positive function W ∈ C ( R d ) and a positivecompact function W such that(1.3) LW := ∞ X i,j =1 ( σσ ∗ ) ij ∂ i ∂ j W + d X i =1 b i ∂ i W ≤ C − W C >
0, then the associated diffusion semigroup has an invariantprobability measure µ with µ ( W ) ≤ C , see for instances [23, 8, 10, 12]. Obviously, thiscondition is not available when b is unbounded on compact sets. Our second purpose isto present a reasonable integrability condition for the existence and uniqueness of invariantprobability measures, which applies to a class of SDEs with locally unbounded coefficients.Moreover, we also intend to investigate the regularity properties of the invariant proba-bility measure. Recall that a probability measure µ on R d is called an invariant probabilitymeasure of the generator L (denoted by L ∗ µ = 0), if(1.4) µ ( Lf ) := Z R d Lf d µ = 0 , f ∈ C ∞ ( R d ) . Obviously, an invariant probability measure µ of the Markov semigroup P t associated to (1.1)satisfies L ∗ µ = 0. In the past two decades, the existence, uniqueness and regularity estimatesfor invariant probability measures of L have been intensively investigated in both finiteand infinite dimensional spaces, see the survey paper [8] for concrete results and historicalremarks. Here, we would like to recall a fundamental result on the regularity of the invariantprobability measures. Let W , loc (d x ) be the class of functions f ∈ L loc (d x ) such that Z R d f ( x )(div G )( x )d x = − Z R d h G, F i ( x )d x, G ∈ C ∞ ( R d → R d )holds for some F ∈ L loc ( R d → R d ; d x ), which is called the weak gradient of f and is denotedby F = ∇ f as in the classical case. For any p ≥
1, let W p, (d x ) = (cid:8) f ∈ W , loc (d x ) : f, |∇ f | ∈ L p (d x ) (cid:9) . Consider the elliptic differential operator L := ∆ + b · ∇ on R d for some locally integrable b : R d → R d . It has been shown in [9] that any invariant probability measure µ of L with µ ( | b | ) := R R d | b | d µ < ∞ has a density ρ := d µ d x such that √ ρ ∈ W , (d x ). In addition,(1.5) Z R d (cid:12)(cid:12) ∇√ ρ (cid:12)(cid:12) d x ≤ Z R d | b | d µ. Since the invariant probability measure µ of L is in general unknown, the integrabilitycondition µ ( | b | ) < ∞ is not explicit. As mentioned above that to ensure the existence of µ one uses the Lyapunov condition (1.3) for some positive function W ∈ C ( R d ) and acompact function W , and to verify µ ( | b | ) < ∞ one would further need | b | ≤ c + cW for some constant c >
0. As we noticed above that these conditions do not apply if thecoefficients merely satisfy an integrability condition with respect to a reference probabilitymeasure.In conclusion, we aim to search for explicit integrability conditions on b and σ with respectto a nice reference measure (for instance, the Gaussian measure) to imply the non-explosionof solutions to the SDE (1.1); the strong Feller property of the associated Markov semigroup;the existence, uniqueness and regularity estimates of the invariant probability measure. Wealso aim to extend the resulting assertions to the infinite-dimensional case.3he main results of this paper will be stated in Section 2. Their proofs are then presentedin Sections 3-6 respectively. Finally, in Section 7 we present a local Harnack inequality whichplays a crucial role in the study.To conclude this section, we present below a simple example to compare our results withexisting ones introduced above. Example 1.1.
Consider, for instance, the following SDE on R d :d X t = { Z ( X t ) − λ X t } d t + √ W t , where λ ∈ R is a constant and Z : R d → R d is measurable.(1) By Theorem 2.1 below for ψ ( x ) = | x | , if(1.6) Z R d e ε | Z ( x ) | − ε − | x | d x < ∞ for some ε ∈ (0 , , then for any initial value the SDE has a unique strong solution which is non-explosive,and the associated Markov semigroup P t is strong Feller with a strictly positive density.Obviously, there are a lot of maps Z satisfying (1.6) but (1.2) and the Lyapunov conditiondoes not hold. For instance, it is the case when(1.7) Z ( x ) := x (cid:26) ∞ X n =1 log(1 + | x − nx | − ) (cid:27) θ for some x ∈ R d with | x | = 1 and θ ∈ (0 , ].(2) When λ >
0, we let µ (d x ) = C e − λ | x | d x be a probability measure with normal-ization constant C >
0. It is well known by Gross [21] that the log-Sobolev inequality inAssumption (H1) holds for κ = λ and β = 0. By Theorem 2.3, if(1.8) Z R d e λ | Z ( x ) | − λ | x | d x < ∞ for some λ > λ , then P t has a unique invariant probability measure µ (d x ) = ρ ( x )d x such that µ (cid:0) |∇√ ρ | (cid:1) ≤ λ λλ − µ (e λ | Z | ) < ∞ ,µ (cid:0) |∇ log ρ (cid:12)(cid:12) ) ≤ µ ( | Z | ) < ∞ . Obviously, for any θ ∈ (0 , ), condition (1.8) holds for Z defined by (1.7), but the Lyapunovcondition (1.3) is not available. In the following four subsections, we introduce the main results in finite-dimensions and theirinfinite-dimensional extensions respectively. To apply integrability conditions with respectto a reference measure µ , we regard the original SDE as a perturbation to the correspondingreference SDE whose semigroup is symmetric in L ( µ ).4 .1 Non-explosion and strong Feller for SDEs Let σ ∈ C ( R d → R d ⊗ R d ) with σ ( x ) invertible for x ∈ R d and denote a = σσ ∗ = ( a ij ) ≤ i,j ≤ d .For V ∈ C ( R d ), define Z = d X i,j =1 { ∂ j a ij − a ij ∂ j V } e i ,L = tr( a ∇ ) + Z · ∇ = d X i,j =1 a ij ∂ i ∂ j + d X i =1 h Z , e i i ∂ i , (2.1)where { e i } di =1 is the canonical orthonormal basis of R d , and ∂ i is the directional derivativealong e i .By the integration by parts formula, L is symmetric in L ( µ ) for µ (d x ) := e − V ( x ) d x : µ ( f L g ) = − µ ( h a ∇ f, ∇ g i ) , f, g ∈ C ∞ ( R d ) . Then E ( f, g ) := µ ( h a ∇ f, ∇ g i ) , f, g ∈ H , σ ( µ )is a symmetric Dirichlet form generated by L , where H , σ ( µ ) is the closure of C ∞ ( R d )under the norm k f k H , σ ( µ ) := (cid:8) µ ( | f | + | σ ∗ ∇ f | ) (cid:9) . When σ ≡ I (the identity matrix), we simply denote H , σ ( µ ) = H , ( µ ).Let W t be the d -dimensional Brownian motion as in Introduction. Consider the referenceSDE(2.2) d X t = Z ( X t )d t + √ σ ( X t )d W t . Since σ and Z are locally Lipschitz continuous, for any initial point x ∈ R d the SDE (2.2)has a unique solution X xt up to the explosion time ζ x . Let P t be the associated (sub-)Markovsemigroup: P t f ( x ) := E (cid:8) { ζ x >t } f ( X xt ) (cid:9) , f ∈ B b ( R d ) , t ≥ , x ∈ R d . When µ (d x ) := e − V ( x ) d x is finite and 1 ∈ H , σ ( µ ) with E (1 ,
1) = 0, we have P t µ -a.e. Since P t t >
0, we have P t x ) = 1 for all t ≥ x ∈ R d . Therefore, in this case the solution to (2.2) is non-explosive for any initialpoints. By the symmetry of P t in L ( µ ), µ is P t -invariant.Now, for a measurable drift Z : R d → R d , we consider the perturbed SDE(2.3) d X t = (cid:8) Z + Z (cid:9) ( X t )d t + √ σ ( X t )d W t . By Itˆo’s formula, the generator of the solution is L := L + Z · ∇ . According to [45, Theorem1.1], if | Z | ∈ L ploc (d x ) for some p > d , then for any initial point x ∈ R d , the SDE (2.3) has aunique solution X xt up to the life time ζ x . We let P t be the associated (Dirichlet) semigroup: P t f ( x ) = E (cid:2) { t<ζ x } f ( X xt ) (cid:3) , x ∈ R d , t ≥ , f ∈ B b ( R d ) . P ( ζ x = ∞ ) = 1 for all x ∈ R d , the solution is called non-explosive. In this case P t is aMarkov semigroup. More generally, for any non-empty open set O ⊂ R d , let T x O = ζ x ∧ inf { t ∈ [0 , ζ x ) : X xt / ∈ O } , inf ∅ := ∞ . Then the associated Dirichlet semigroup on O is given by P O t f ( x ) = E (cid:2) { t Theorem 2.1. Let σ ∈ C ( R d → R d ⊗ R d ) with σ ( x ) invertible for x ∈ R d , and let V ∈ C ( R d ) such that (2.4) Z R d (cid:16) | σ ∗ ∇ ψ ( x ) | + e ε | ( σ − Z )( x ) | (cid:17) e − V ( x ) − ε − ρ σ (0 ,x ) d x < ∞ holds for some constant ε ∈ (0 , and a local Lipschitz continuous compact function ψ on R d . Then (2.3) has a unique non-explosive solution for any initial points, and the associatedMarkov semigroup P t is strong Feller with at most one invariant probability measure. More-over, for any non-empty open set O ⊂ R d and t > , P O t is strong Feller and has a strictlypositive density p O t with respect to the Lebesgue measure on O . Remark 2.1. (1) Typical choices of ψ include | x | , log(1+ | x | ) , log log(e+ | x | ) ... For instance,with ψ ( x ) := log log(e+ | x | ) one may replace the term | σ ∗ ∇ ψ ( x ) | in (2.4) by k σ ( x ) k (e+ | x | ) { log(e+ | x | ) } . So, if V = 0 and R R d e − λρ σ (0 , · ) d x < ∞ for some λ > 0, the condition (2.4) holds providedlog k σ k (e + | · | ) { log(e + | · | ) } + | σ − Z | ≤ C (1 + ρ σ (0 , · )) + f for some constant C > f with e εf − ε − ρ σ (0 , · ) ∈ L (d x ) for some ε ∈ (0 , σ ( r ) = sup | x | = r k σ ( x ) k for r ≥ 0. Then ρ σ (0 , x ) ≥ U ( x ) := Z | x | d r ¯ σ ( r ) , x ∈ R d . So, in (2.4) we may replace ρ σ (0 , · ) by the more explicit function U .(3) The condition σ ∈ C ( R d → R d ⊗ R d ) is stronger than σ ∈ W p, loc ( R d → R d ⊗ R d ; d x )for some p > σ implies the local Harnack inequality (see Theorem 7.1 below), which is a crucial toolin our study. If the local Harnack inequality could be established under weaker conditions,6his condition would be weakened automatically. Indeed, under an additional assumption,this condition will be replaced by σ ∈ W p, loc ( R d → R d ⊗ R d ; d x ) for some p > 1, see Theorem2.4 below for details.Intuitively, the non-explosion is a long distance property of the solution. So, it is naturalfor us to weaken the integrability condition (2.4) by taking the integral outside a compactset. But under this weaker condition we are not able to prove other properties included inTheorem 2.1. Corollary 2.2. Let σ ∈ C ( R d → R d ⊗ R d ) with σ ( x ) invertible for x ∈ R d , and let V ∈ C ( R d ) such that (2.5) Z D c (cid:16) | σ ∗ ∇ ψ ( x ) | + e ε | ( σ − Z )( x ) | (cid:17) e − V ( x ) − ε − ρ σ (0 ,x ) d x < ∞ holds for some compact set D ⊂ R d , some constant ε ∈ (0 , , and some local Lipschitzcontinuous compact function ψ on R d . If Z ∈ L ploc (d x ) for some constant p > d , then theSDE (2.3) has a unique non-explosive solution for any initial points. To investigate the invariant probability measures for the SDE (2.3), we need the non-explosion of solutions such that the standard tightness argument for the existence of in-variant probability measure applies. To this end, we will apply Theorem 2.1 above, forwhich we first assume that σ is C -smooth (see (H1) below) then extend to less regular σ by approximations (see (H ′ ) below). Assumption (H1) (1) σ ∈ C ( R d → R d ⊗ R d ) with σ ( x ) invertible for x ∈ R d , V ∈ C ( R d ) such that µ (d x ) := e − V ( x ) d x is a probability measure satisfying(2.6) H , σ ( µ ) = W , σ ( µ ) := (cid:8) f ∈ W , loc (d x ) : f, | σ ∗ ∇ f | ∈ L ( µ ) (cid:9) . (2) The (defective) log-Sobolev inequality(2.7) µ ( f log f ) ≤ κµ ( | σ ∗ ∇ f | ) + β, f ∈ C ∞ ( R d ) , µ ( f ) = 1holds for some constants κ > , β ≥ . Since µ (d x ) := e − V ( x ) d x is finite, (2.6) implies 1 ∈ H , σ ( µ ) with E (1 , 1) = 0, so that thesolution to (2.2) is non-explosive as explained above. We note that (2.6) holds if the metric ρ σ is complete. Indeed, in this case the function ρ σ (0 , · ) is compact with | σ ∗ ∇ ρ σ (0 , · ) | = 1,so that for any f ∈ W , σ ( µ ) we have f n := f { ∧ ( n + 1 − ρ σ (0 , · )) + } ∈ H , σ ( µ ) for n ≥ f n → f in the norm k · k H , σ ( µ ) . σσ ∗ ≥ αI and Hess V ≥ KI for some constants α, K > 0, then the Bakry-Emerycriterion [5] implies (2.7) for κ = Kα . In the case that K is not positive, the log-Sobolevinequality holds for some constant κ > µ (e λ |·| ) < ∞ for some ε > − K , see [36, Theorem1.1]. See also [13] for a Lyapunov type sufficient condition of the log-Sobolev inequality. Theorem 2.3. Assume (H1) and that (2.8) µ (e λ | σ − Z | ) := Z R d e λ | σ − Z | d µ < ∞ holds for some constant λ > κ . Let P t be the semigroup associated to (2.3) , and let L = L + Z · ∇ for L in (2.1) . Then: (1) L has an invariant probability measure µ , which is absolutely continuous with respect to µ such that the density function ρ := d µ d µ is strictly positive with √ ρ, log ρ ∈ H , σ ( µ ) and (2.9) µ (cid:0) | σ ∗ ∇√ ρ | (cid:1) ≤ λ − κ (cid:8) log µ (e λ | σ − Z | ) + β (cid:9) < ∞ ;(2.10) µ ( | σ ∗ ∇ log ρ | ) := lim δ ↓ Z R d | σ ∗ ∇ ρ | ( ρ + δ ) d µ ≤ µ ( | σ − Z | ) < ∞ . (2) The measure µ is the unique invariant probability measure of L and P t provided (2.11) µ (e ε k σ k ) < ∞ for some constant ε > . Remark 2.2. (1) Simply consider the case that σ = σ = I . If Hess V ≥ K for some K > 0, then (H1) holds for κ = K and β = 0. So, when µ (e λ | Z | ) < ∞ holds for some λ > K , Theorem 2.3 implies that L and P t have a unique invariant probability measure µ ,which is absolutely continuous with respect to µ , and the density function satisfies ρ := d µ d µ satisfies √ ρ, log ρ ∈ H , ( µ ) with µ ( |∇√ ρ | ) ≤ K Kλ − µ (e λ | Z | ) < ∞ ; µ ( |∇ log ρ | ) ≤ µ ( | Z | ) < ∞ . (2) Under (H1) , if the super log-Sobolev inequality µ ( f log f ) ≤ rµ ( | σ ∗ ∇ f | ) + β ( r ) , r > , f ∈ C ∞ ( R d ) , µ ( f ) = 1holds for some β : (0 , ∞ ) → (0 , ∞ ), then Theorem 2.3 applies when (2.8) holds for some λ > 0, and in this case (2.9) reduces to µ (cid:0) | σ ∗ ∇√ ρ | (cid:1) ≤ inf λ> ,r ∈ (0 , λ ) λ − r (cid:8) log µ (e λ | σ − Z | ) + β ( r ) (cid:9) < ∞ . M = R d , the super log-Sobolevinequality holds provided a ≥ αI for some constant α > V is bounded below with µ (e λ |·| ) < ∞ for any λ > 0. In particular, it is the case when a = I, V ( x ) = c + c | x | p forsome constants c ∈ R , c > p > 2. See [18, 21, 35] and references within for morediscussions on the super log-Sobolev inequality and the corresponding semigroup property.(3) To illustrate the sharpness of condition (2.8) for some λ > κ , let us consider σ = σ = I and V ( x ) = c + | x | for some constant c ∈ R , so that (H1) holds for κ = 2 and β = 0. Let Z ( x ) = rx = r ∇| · | ( x ) for some constant r ≥ 0. It is trivial that L has aninvariant probability measure if and only if r < , which is equivalent to µ (e λ | Z | ) < ∞ forsome λ > κ = . Now, we extend Theorem 2.3 to less regular σ by using the following assumption toreplace (H1) . Assumption (H ′ ) (1) σ ∈ W p, loc ( R d → R d ⊗ R d ; d x ) for some p > d , σ ( x ) is invertible for every x ∈ R d , and a := σσ ∗ ≥ αI for some constants α > V ∈ C ( R d ) such that µ (d x ) := e − V ( x ) d x is a probability measure satisfying (2.6) and(2.12) µ ( f log f ) ≤ κ ′ µ ( |∇ f | ) + β, f ∈ C ∞ ( R d ) , µ ( f ) = 1for some constants κ ′ > , β ≥ . (3) There exists a constant p > a ij ∈ H , ( µ ) ∩ L p ( µ ) for any 1 ≤ i, j ≤ d and |∇ V | ∈ L pp − ( µ ) . Let L and P t be in Theorem 2.3 associated to the SDE (2.3). Theorem 2.4. Assume (H ′ ) and let µ (cid:0) exp[ λ | Z | ] (cid:1) < ∞ hold for some λ > κ ′ α . Then L and P t have a unique invariant probability measure µ (d x ) := ρ ( x ) µ (d x ) for some strictlypositive function ρ such that √ ρ, log ρ ∈ H , ( µ ) with (2.13) µ (cid:0) |∇√ ρ | (cid:1) ≤ α λ − κ ′ (cid:8) log µ (cid:0) e λ | Z | (cid:1) + β (cid:9) < ∞ ;(2.14) µ ( |∇ log ρ | ) := lim δ ↓ Z R d |∇ ρ | ( ρ + δ ) d µ ≤ α µ ( | Z | ) < ∞ . We first consider the invariant probability measure of second order differential operators on aseparable Hilbert space, then apply to semi-linear SPDEs. We will take a Gaussian measureas the reference measure. 9et ( H , h· , ·i , | · | ) be a separable Hilbert space, let ( A, D ( A )) be a positive definite self-adjoint operator on H having discrete spectrum with all eigenvalues (0 < ) λ ≤ λ ≤ · · · counting multiplicities such that(2.15) ∞ X i =1 λ − i < ∞ . Let { e i } i ≥ be the corresponding eigenbasis of A . Let µ be the Gaussian measure on H withcovariance operator A − . In coordinates with respect to the basis { e i } i ≥ , we have(2.16) µ (d x ) = ∞ Y i =1 (cid:16) √ λ i √ π e − λix i d x i (cid:17) , x i := h x, e i i , i ≥ . For any n ≥ 1, let H n = span { e i : 1 ≤ i ≤ n } and define the probability measure µ ( n )0 (d x ) = n Y i =1 (cid:16) √ λ i √ π e − λix i d x i (cid:17) on H n . We have µ ( n )0 = µ ◦ π − n for the orthogonal projection π n : H → H n .Let ( L ( H ) , k · k ) be the space of bounded linear operators on H with operator norm k · k , and let L s ( H ) be the class of all symmetric elements in L ( H ). For any a ∈ L s ( H ) let a ij = h ae i , e j i for i, j ≥ 1. We make the following assumption. Assumption (H2) (1) a ij ∈ C ( H ) for i, j ≥ 1, and a ≥ αI for some constant α > n ≥ σ n := p ( a ij ) ≤ i,j ≤ n , H , σ n ( µ ( n )0 ) = W , σ n ( µ ( n )0 ) holds.(3) For any i, j ≥ 1, there exists ε ij ∈ (0 , 1) such that(2.17) sup n ≥ Z R n exp (cid:2) ε ij | a ij | ε ij (cid:3) d µ ( n )0 < ∞ . We note that Z R n exp (cid:2) ε ij | a ij | ε ij (cid:3) d µ ( n )0 = Z H exp (cid:2) ε ij | a ij ◦ π n | ε ij (cid:3) d µ . As mentioned above that H , σ n ( µ ( n )0 ) = W , σ n ( µ ( n )0 ) is implied by the completeness of themetric on R n induced by σ n , and the later holds if for any i, j ≥ ε ij > | a ij ( x ) | ≤ ε ij (1 + | x | ) , x ∈ H . µ ( n )0 , the conditions(2.17) and (2.18) hold provided for any i, j ≥ ε ′ ij ∈ (0 , 1) such that | a ij ( x ) | ≤ ε ′ ij (1 + | x | ) ε ′ ij . Let ∂ i be the directional derivative along e i , i ≥ . For a measurable drift Z : H → H ,consider the operators(2.19) L := L + Z · ∇ , L := ∞ X i,j =1 (cid:16) a ij ∂ i ∂ j + (cid:8) ∂ j a ij − a ij λ j (cid:9) ∂ i (cid:17) , which are well defined on the class of smooth cylindrical functions with compact support: F C ∞ := (cid:8) H ∋ x f ( h x, e i , · · · , h x, e n i ) : n ≥ , f ∈ C ∞ ( R n ) (cid:9) . It is easy to see that L is symmetric in L ( µ ): µ ( f L g ) = − µ ( h a ∇ f, ∇ g i ) , f, g ∈ F C ∞ . Let H , ( µ ) be the completion of F C ∞ with respect to the inner product h f, g i H , ( µ ) := µ ( f g ) + µ ( h∇ f, ∇ g i ) . A probability measure µ on H is called an invariant probability measure of L (denoted by L ∗ µ = 0), if for any f ∈ F C ∞ we have Lf ∈ L ( µ ) and µ ( Lf ) = 0. Theorem 2.5. Assume (2.15) and (H2) . (1) If µ (e λ | Z | ) < ∞ for some λ > λ α , then L has an invariant probability measure µ ,which is absolutely continuous with respect to µ , and the density function ρ := d µ d µ satisfies √ ρ, log ρ ∈ H , ( µ ) with (2.20) µ (cid:0)(cid:12)(cid:12) ∇√ ρ (cid:12)(cid:12) (cid:1) ≤ λ α λ λ − µ (e λ | Z | ) < ∞ and (2.21) µ (cid:0)(cid:12)(cid:12) ∇ log ρ (cid:12)(cid:12) (cid:1) := lim δ ↓ µ (cid:0)(cid:12)(cid:12) ∇ log ρ + δ (cid:12)(cid:12) (cid:1) ≤ µ ( | Z | ) α < ∞ . (2) If moreover k a k ∞ < ∞ , then ( L, F C ∞ ) is closable in L ( µ ) and the closure generatesa Markov C -semigroup T t on L ( µ ) with µ as an invariant probability. Moreover,there exists a standard Markov process { ¯ P x } x ∈ H ∪{ ∂ } on H ∪ { ∂ } which is continuousand non-explosive for E µ -q.e. x , such that the associated Markov semigroup ¯ P t is a µ -version of T t ; that is, ¯ P t f = T t f µ -a.e. for all t ≥ and f ∈ B b ( H ) . ∂ be an extra point and extend the topology of H to H ∪ { ∂ } by letting the set { ∂ } open. A family of probability measures { P x } x ∈ H ∪{ ∂ } onΩ := (cid:8) ω : [0 , ∞ ) → H ∪ { ∂ } : if ω t = ∂ then ω s = ∂ for s ≥ t (cid:9) equipped with the σ -field F := σ ( ω t : t ≥ 0) is called a standard Markov process, if P x ( ω = x ) = 1 and the distribution P t ( x, d y ) of Ω ∋ ω ω t under P x gives rise to aMarkov transition kernel on H ∪ { ∂ } . When the process is non-explosive, i.e. P x (cid:0) inf { t ≥ ω t = ∂ } = ∞ (cid:1) = 1 , x ∈ H , the sub-family { P x } x ∈ H is a standard Markov process on H . In this case, the process (or theassociated Markov semigroup P t ) is called Feller if P t C b ( H ) ⊂ C b ( H ) for all t ≥ 0, and iscalled strong Feller if P t B b ( H ) ⊂ C b ( H ) for all t > . If moreover P x ( C ([0 , ∞ ) → H )) = 1holds for all x ∈ H , then the process is continuous.Next, we extend Theorem 2.4 to the infinite-dimensional case, for which we need thefollowing assumption. Assumption (H ′ ) (1) a ≥ αI for some constant α > 0, and for every n ≥ p > n such that a n ∈ W p, loc ( R n → R n ⊗ R n ; d x ) . (2) For any i, j ∈ N there exists ε ij ∈ (0 , 1) such that (2.17) and µ ( n )0 ( |∇ a ij ◦ π n | ε ij ) < ∞ hold for any n ≥ Theorem 2.6. Under (2.15) and (H ′ ) , assertions (1) and (2) in Theorem . hold. We intend to investigate the existence, uniqueness and non-explosion of the SPDE corre-sponding to L in (2.19), and to show that the probability measure in Theorem 2.5 is theunique invariant probability measure of the associated Markov semigroup. For technicalreasons, we only consider the case that a = I , for which the corresponding SPDE reduces tothe standard semi-linear SPDE(2.22) d X t = (cid:8) Z ( X t ) − AX t (cid:9) d t + √ W t , where Z : H → H is measurable, W t is the cylindrical Brownian motion, i.e. W t = ∞ X i =1 β it e i , t ≥ { β it } i ≥ . An adapted con-tinuous process X t on H is called a mild solution to (2.22), if X t = e − At X + Z t e − A ( t − s ) Z ( X s )d s + Z t e − ( t − s ) A d W s , t ≥ . We assume (H3) P ∞ i =1 1 λ θi < ∞ for some θ ∈ (0 , µ (e λ | Z | ) < ∞ for some constant λ > (H3) implies the existence and pathwise uniquenessof mild solutions to (2.22) for µ -a.e. starting points. Below we intend to prove the weakuniqueness of (2.22) for any initial points. A standard continuous Markov process on H is called a weak solution to (2.22), if it solves the martingale problem for ( L, F C ∞ ). Inthis case one may construct a cylindrical Brownian motion W t on the probability space( C ([0 , ∞ ) → H ); F , P x ), where F := σ ( { ω ω t : t ≥ } ), such that the coordinate process X t ( ω ) := ω t is a mild solution to (2.22) with X = x . See e.g. [24, Proposition IV.2.1] forthe explanation in the finite-dimensional case, which works also in the present case as thecylindrical Brownian motion is determined by its finite-dimensional projections. Theorem 2.7. Assume that (H3) holds. (1) There exists a standard continuous Markov process { P x } x ∈ H solving (2.22) weakly forevery initial point, and the associated Markov semigroup P t is strong Feller having astrictly positive density with respect to µ . (2) If Z is bounded on bounded sets, then there exists a unique standard Markov processsolving (2.22) weakly for every initial point such that the associated Markov semigroupis Feller. (3) If Z is bounded on bounded sets and µ (e λ | Z | ) < ∞ holds for some λ > λ , then P t hasa unique invariant probability measure µ , which is absolutely continuous with respectto µ and the density function ρ := d µ d µ is strictly positive with √ ρ, log ρ ∈ H , ( µ ) such that estimates (2.20) and (2.21) hold for α = 1 . Remark 2.3. Unlike in the finite-dimensional case where Z ∈ L ploc (d x ) for some p >d implies the pathwise uniqueness of the solution for any initial points, in the infinite-dimensional case this is unknown without any continuity conditions on Z . It is shown in[39] (also for the multiplicative noise case) that if Z is Dini continuous then the pathwiseuniqueness holds for any initial points. The main idea is to show that the solution to the reference SDE (2.2) is a weak solution to(2.3) under a weighted probability, so that the non-explosion of (2.2) implies that of (2.3).13o this end, we will apply the local Harnack inequality (3.2) below to verify the Novikovcondition for the Girsanov transform. To realize the idea, we first consider the case that(3.1) Z R d e ε | ( σ − Z )( x ) | − V ( x ) d x < ∞ holds for some ε > 0, then reduce back to the original condition (2.4). Lemma 3.1. Assume that (3.1) holds for some constant ε > and ∈ H , σ ( µ ) with E (1 , 1) = 0 , then all assertions in Theorem . hold.Proof. Obviously, (3.1) implies that µ (d x ) := e − V ( x ) d x is a finite measure. Since the co-efficients in (2.2) is locally Lipschitz continuous, it is classical that the SDE has a uniquesolution up to the explosion time. Since 1 ∈ H , σ ( µ ) with E (1 , 1) = 0, as explained after(2.2) that the solution to (2.2) is non-explosive and µ is P t -invariant. Moreover, since thedrift in (2.3) is locally bounded, according to [44], this SDE has a unique solution for anyinitial points. So, it remains to show that the solution is non-explosive, and the associatedMarkov semigroup P t is strong Feller with at most one invariant probability measure.A crucial tool in the proof is the following local Harnack inequality. Consider R d withthe C -Riemannian metric h u, v i σ := h σσ ∗ u, v i , u, v ∈ R d , and let ∆ σ , ∇ σ be the corresponding Laplace-Beltrami operator and the gradient operator.Then L can be rewritten as L = ∆ σ + ∇ σ ¯ V for some ¯ V ∈ C ( R d ) . Since the intrinsic distance ρ σ is locally equivalent to the Euclideandistance, according to Theorem 7.1 below, for any p > p ∈ C ( R d )such that(3.2) ( P t f ) p ( x ) ≤ ( P t f p ( y )) exp (cid:20) Φ p ( x ) (cid:16) | x − y | ∧ t (cid:17)(cid:21) x, y ∈ R d , | x − y | ≤ p ( x )holds for all t > f ∈ B + b ( R d ) := { f ∈ B b ( R d ) : f ≥ } . (a) Non-explosion . It suffices to find out a constant t > t . To this end, we construct a weaksolution by using the reference SDE (2.2). We intend to find out t > x , the solution to (2.2) for X = x is a weak solution to (2.3) for t ∈ [0 , t ] . So,by the weak uniqueness of (2.3), which follows from the strong uniqueness, we conclude thatthe strong solution to (2.3) is non-explosive before t . To this end, we verify the Novikovcondition(3.3) E exp (cid:20) Z t | ( σ − Z )( X s ) | d s (cid:21) < ∞ , X = x ∈ R d , 14o that Q := exp[ √ R t h ( σ − Z )( X s ) , d W s i − R t | ( σ − Z )( X s ) | ] P is a probability measure.In this case, by the Girsanov theorem,˜ W t := W t − √ Z t ( σ − Z )( X s ) d s, t ∈ [0 , t ]is a Brownian motion under Q . Thus, rewriting (2.2) asd X t = ( Z + Z )( X t ) + √ σ ( X t )d ˜ W t , t ∈ [0 , t ] , we see that ( X t , ˜ W t ) t ∈ [0 ,t ] is a weak solution to (2.3) under the probability measure Q .To prove (3.3), we use the Harnack inequality (3.2) for p = d + 1 to derive (cid:8) E e λ ( | ( σ − Z )( X t ) | ∧ N ) (cid:9) d +1 = (cid:0) P t e λ ( | σ − Z | ∧ N ) ( x ) (cid:1) d +1 ≤ P t e ( d +1) λ ( | σ − Z | ∧ N ) ( y )e Φ d +1 ( x )(1+ | x − y | /t ) , t ∈ (0 , , N > , | y − x | ≤ d +1 ( x ) . Since µ is P t -invariant, for B x,t := (cid:8) y : | y − x | ≤ d +1 ( x ) ∧ √ t (cid:9) this implies (cid:8) E exp (cid:2) λ ( | ( σ − Z )( X t ) | ∧ N ) (cid:3)(cid:9) d +1 µ ( B x,t )e − d +1 ( x ) ≤ Z B x,t (cid:0) P t e λ ( | σ − Z | ∧ N ) (cid:1) d +1 ( x ) exp h − Φ d +1 ( x ) (cid:16) | x − y | t (cid:17)i µ (d y ) ≤ Z B x,t P t e ( d +1) λ ( | σ − Z | ∧ N ) ( y ) µ (d y ) ≤ µ (e ε | σ − Z | ) < ∞ , t ∈ (0 , , λ ∈ (cid:16) , εd + 1 i . Since µ has strictly positive and continuous density e − V with respect to d x , there exists G ∈ C ( R d → (0 , ∞ )) such that µ ( B x,t ) ≥ G ( x ) t d for t ∈ (0 , 1] and x ∈ R d . By taking λ = ε/ ( d + 1) and letting N → ∞ in the above display, we arrive at E e εd +1 | ( σ − Z )( X t ) | ≤ H ( x ) √ t < ∞ , t ∈ (0 , , x ∈ R d for some positive H ∈ C ( R d ) . Therefore, by Jensen’s inequality, we have E exp (cid:20) γ Z r | ( σ − Z )( X s ) | d s (cid:21) ≤ r Z r E e γr | ( σ − Z )( X s ) | d s ≤ r Z r H ( x ) √ t d t = 2 H ( x ) √ r < ∞ , x ∈ R d , r ∈ (0 , , γ ∈ (cid:16) , ε ( d + 1) r i . (3.4)This implies (3.3) by taking γ = and t = r = 1 ∧ εd +1 . (b) Strong Feller of P t and uniqueness of invariant probability measure . Ac-cording to [7, Theorem 4.1], the Markov semigroup P t is strong Feller. For any x ∈ R d , welet X xs solve (2.2) with initial point x and define R xr = exp (cid:20) √ Z r h ( σ − Z )( X xs ) , d W s i − Z r | ( σ − Z )( X xs ) | d s (cid:21) , r ∈ [0 , t ] . 15y (3.3) and the Girsanov theorem, we have P t f ( x ) = E (cid:2) f ( X xt ) R xt (cid:3) , t ∈ [0 , t ] , f ∈ B b ( R d ) , x ∈ R d . Then for any t > , x ∈ R d and f ∈ B b ( R d ), the semigroup property of P s and the strongFeller property of P s implylim sup y → x | P t f ( y ) − P t f ( x ) | = lim sup y → x | P r ( P t − r f )( y ) − P r ( P t − r f )( x ) | = lim sup y → x (cid:12)(cid:12) E [ R yr P t − r f )( X yr ) − R xr ( P t − r f )( X xr )] (cid:12)(cid:12) ≤ lim sup y → x n(cid:12)(cid:12) P r ( P t − r f )( y ) − P r ( P t − r f )( x ) (cid:12)(cid:12) + E (cid:0) | R yr − | + | R xr − | (cid:1)o ≤ sup y : | y − x |≤ E (cid:0) | R yr − | + | R xr − | (cid:1) , r ∈ (0 , t ) . Noting that E | R yr − | = E ( R yr ) − r > 0, then the strong Feller property followsprovided(3.5) lim sup r → sup y : | y − x |≤ E ( R yr ) ≤ . To prove this, we let M r = √ R r h ( σ − Z )( X xs ) , d W s i . Then for small r > E ( R yr ) = E e M r −h M i r ≤ (cid:0) E e M r − h M i r (cid:1) (cid:0) E e h M i r (cid:1) = (cid:0) E e R r | ( σ − Z )( X xs ) | d s (cid:1) . So, applying (3.4) with γ = ε ( d +1) r for small r > 0, and using Jensen’s inequality, we obtainlim sup r → sup y : | y − x |≤ (cid:8) E ( R yr ) (cid:9) ≤ lim sup r → sup y : | y − x |≤ E e R r | ( σ − Z )( X xs ) | d s ≤ lim sup r → sup y : | y − x |≤ (cid:0) E e γ R r | ( σ − Z )( X xs ) | d s (cid:1) γ ≤ lim sup r → sup y : | y − x |≤ (cid:16) H ( y ) √ r (cid:17) d +1) rε = 1 . This implies (3.5).Next, as already mentioned above, every invariant probability measure of P t has strictlypositive density with respect to the Lebesgue measure, so that any two invariant probabilitymeasures are equivalent each other. Therefore, the invariant probability measure has to beunique, since it is well known that any two different extremal invariant probability measuresof a strong Feller Markov operator are singular each other.(c) The assertion for P O t . Due to the semigroup property ensured by the pathwiseuniqueness, it suffices to prove for small enough t > 0. Let T x O be the hitting time of X xt tothe boundary of O . By the Girsanov theorem we have(3.6) P O t f ( x ) = E (cid:2) { T x O >t } f ( X xt ) R xt (cid:3) , f ∈ B b ( O ) , x ∈ O . Let P O , t f ( x ) = E (cid:2) { T x O >t } f ( X xt ) (cid:3) be the Dirichlet semigroup associated to (2.3). Since σ isinvertible, by the C -regularity of σ and V we see that P O , t is strong Feller having strictly16ositive density with respect to the Lebesgue measure (see [4] for gradient estimates andlog-Harnack inequalities of P O , t ). Then the strong Feller property can be proved as in (b)using P O , t in place of P t .Next, by (3.4) we have E { ( R xt ) − } < ∞ for small t > 0. Then for any measurable set A such that P O t A ( x ) = 0, (3.6) implies { P O , t A ( x ) } = (cid:8) E (cid:2) { T x O >t } A ( X xt ) (cid:3)(cid:9) ≤ { P O t A ( x ) } E { ( R xt ) − } = 0 . Thus, the measure P O , t d z ( x ) is absolutely continuous to the measure P O t d z ( x ) . Since P O , t has a strictly positive density, so does P O t . Proof of Theorem 2.1. Since | σ ∗ ∇ ρ σ (0 , · ) | = 1, for any δ > ρ σ (0 , · ) can beuniformly approximated by smooth ones f n with | σ ∗ ∇ f n | ≤ δ . In particular, we maytake ˜ ρ ∈ C ( R d ) such that | ρ σ (0 , · ) − ˜ ρ | ≤ | σ ∗ ∇ ˜ ρ | ≤ 2, so that (2.4) holds for some ε ∈ (0 , 1) if and only if(3.7) Z R d (cid:16) | σ ∗ ∇ ψ ( x ) | + e ε | ( σ − Z )( x ) | (cid:17) e − V ( x ) − ε − ˜ ρ ( x ) d x < ∞ holds for some ε ∈ (0 , . To apply Lemma 3.1, we take¯ µ (d x ) := e − V ( x ) − ε − ˜ ρ ( x ) d x R R d e − V ( x ) − ε − ˜ ρ ( x ) d x , which is a probability measure by (3.7). Let¯ Z ( x ) = Z ( x ) − ε − a ( x ) ∇ ˜ ρ ( x ) , ¯ Z ( x ) = Z ( x ) + 2 ε − a ( x ) ∇ ˜ ρ ( x ) . By (3.7) we have ¯ µ ( | σ ∗ ∇ ψ | ) < ∞ , so that f n := ( n − ψ ) + ∧ → L (¯ µ ) andlim n →∞ ¯ µ ( | σ ∗ ∇ f n | ) = lim n →∞ Z n ≥ ψ ≥ n | σ ∗ ∇ ψ | d¯ µ = 0 . Thus, 1 ∈ H , σ (¯ µ ) and ¯ E (1 , 1) = 0 . Then by Lemma 3.1 for ( ¯ Z , ¯ Z, ¯ µ ) in place of ( Z , Z, µ ),and due to (3.7), it remains to prove ¯ µ (e ε ′ | σ − ¯ Z | ) < ∞ for some ε ′ > . Since | σ ∗ ∇ ˜ ρ | ≤ | σ − ¯ Z | ( x ) ≤ | σ − Z | ( x ) + 8 ε − | ( σ ∗ ∇ ˜ ρ ( x ) | ( x ) ≤ | σ − Z | ( x ) + 64 ε − ˜ ρ ( x ) . By (2.4), for ε ′ ∈ (0 , ε ] we have¯ µ (e ε ′ | σ − ¯ Z | ) ≤ R R d e − V ( x ) − ε − ˜ ρ ( x ) d x Z R d e ε ′ | σ − Z | ( x )+64 ε ′ ε − ˜ ρ ( x ) − V ( x ) − ε − ˜ ρ ( x ) d x ≤ R R d e − V ( x ) − ε − ˜ ρ ( x ) d x Z R d e ε | σ − Z | ( x ) − V ( x ) − ε − ˜ ρ ( x ) d x < ∞ . Therefore, the proof is finished. 17 roof of Corollary . . Let n ≥ B n := {| · | ≤ n } ⊃ D . It suffices to show thatfor any l ≥ n + 1 and any x ∈ S l := {| · | = l } , the solution ¯ X xt to (2.3) is non-explosive. Let ζ x = lim m →∞ inf { t ≥ | ¯ X xt | ≥ m } , σ xn = inf { t ≥ | ¯ X t | ≤ n } , m > l ≥ n + 1 , x ∈ S l . Let ˜ X xt solve the SDE (2.3) for Z B cn in place of Z . Due to (2.5), Theorem 2.1 applies to˜ X t . In particular, ˜ X xt is non-explosive, i.e.(3.8) ˜ ζ x := lim m →∞ inf { t ≥ | ˜ X xt | ≥ m } = ∞ , where and in the following, inf ∅ := ∞ . Moreover, since | Z | ∈ L ploc (d x ) for some p > d , [45,Theorem 1.1] implies the pathwise uniqueness of the SDE (2.3). So,˜ X xt = ¯ X xt , t ≤ σ xn . Then(3.9) σ xn = ˜ σ xn := inf { t ≥ | ˜ X xt | ≤ n } and(3.10) ζ x = ˜ ζ x if ζ x ≤ σ xn . Obviously, for θ xn := inf { t ≥ σ xn : | X xt | ≥ l } we have(3.11) { σ xn < ζ x } = { θ xn < ζ x } . By (3.8), (3.11) and the strong Markov property ensured by the uniqueness (see [24, Theorem5.1]), we have P ( ζ x ≤ T ) = P ( ζ x ≤ T, σ xn ≥ ζ x ) + P ( ζ x ≤ T, σ xn < ζ x ) ≤ P ( ˜ ζ x ≤ T ) + P ( θ xn < ζ x ≤ T ) = E (cid:2) { θ xn ≤ T } P ( θ xn < ζ x ≤ T | F θ xn ) (cid:3) = E (cid:2) { θ xn ≤ T } { P ( ζ z ≤ T − s ) | s = θ xn ,z = X xθxn } (cid:3) ≤ P ( θ xn ≤ T ) sup z ∈ S l P ( ζ z ≤ T ) ≤ P ( σ xn ≤ T ) sup z ∈ S l P ( ζ z ≤ T ) , T > , x ∈ S l . Combining this with (3.9) we obtain(3.12) sup x ∈ S l P ( ζ x ≤ T ) ≤ n sup x ∈ S l P (˜ σ xn ≤ T ) o sup x ∈ S l P ( ζ x ≤ T ) , T > . Let ˜ P O t be the Dirichlet semigroup of ˜ X · t for O = B cn . By applying Theorem 2.1 for Z B cn inplace of Z , we obtain P (˜ σ xn ≤ T ) = 1 − P (˜ σ xn > T ) = 1 − ˜ P O T x ) < P (˜ σ xn ≤ T ) is continuous in x ∈ O . So, ε T := sup x ∈ S l P ( σ xn ≤ T ) < . This together with (3.12) implies P ( ζ x ≤ T ) = 0 for any T > x ∈ S l . Since l ≥ n + 1is arbitrary and the solution is continuous, we have P ( ζ x = ∞ ) = 1 for all x ∈ R d .18 Proofs of Theorem 2.3 and Theorem 2.4 Since the uniqueness of invariant probability measure is ensured by the irreducibility and thestrong Feller property, we only prove the existence and regularity estimates on the density.The new technique in the proof of the existence is to reduce the usual tightness conditionto the boundedness of a Feyman-Kac semigroup, which follows from the hypercontractivityof P t under the given integrability condition. Moreover, to estimate the derivative of thedensity, the formula (4.3) below will play a crucial role. Lemma 4.1. Let V ∈ W , loc (d x ) and σ ∈ W , loc ( R d → R d ⊗ R d ; d x ) such that µ (d x ) =e − V ( x ) d x is a probability measure satisfying (2.6) and the Poincar´e inequality (4.1) µ ( f ) ≤ Cµ ( | σ ∗ ∇ f | ) + µ ( f ) , f ∈ C ∞ ( R d ) for some constant C > . Let L be in (2.1) and let L := L + Z · ∇ for some measurable Z : R d → R d . If Z has compact support and | Z | + |∇ σ | ∈ L p (d x ) for some p ∈ [2 , ∞ ) ∩ ( d, ∞ ) ,then any invariant probability measure µ of L is absolutely continuous with respect to µ withdensity ρ := d µ d µ ∈ H , σ ( µ ) satisfying (4.2) µ ( ρ + | σ ∗ ∇ ρ | ) ≤ ( C + 1) µ ( ρ | σ ∗ Z | ) < ∞ . Moreover, (4.3) Z R d h σ ∗ ∇ f, σ ∗ ∇ ρ i d µ = Z R d h Z, ∇ f i d µ, f ∈ H , σ ( µ ) . Proof. Let µ be an invariant probability measure of L . Since | Z | + |∇ σ | is in L ploc (d x ) forsome p ∈ [2 , ∞ ) ∩ ( d, ∞ ), by the local boundedness of Z so is | Z + Z | . Then accordingto [8, Corollary 1.2.8], for any invariant probability measure µ of L , µ (d x ) = ˆ ρ ( x )d x holdsfor some ˆ ρ ∈ W p, loc (d x ). Since µ (d x ) = e − V ( x ) d x and V ∈ C ( R d ), this implies µ = ρµ forsome ρ ∈ W , loc (d x ) . In particular, we may take a continuous version ρ which is thus locallybounded. By the integration by parts formula, Z R d h σ ∗ ∇ ρ, σ ∗ ∇ f i d µ = − Z R d ρL f d µ = − Z R d Lf d µ + Z R d h Z, ∇ f i d µ = Z R d h σ − Z, σ ∗ ∇ f i ρ d µ , f ∈ C ∞ ( R d ) . (4.4)Since Z has compact support with | Z | ∈ L (d x ), and ρ + k σ − k is locally bounded, (4.4)implies (cid:12)(cid:12)(cid:12)(cid:12) Z R d h σ ∗ ∇ ρ, σ ∗ ∇ f i d µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ( ρ | σ − Z | ) µ ( | σ ∗ ∇ f | ) < ∞ , f ∈ C ∞ ( R d ) . Hence, µ ( | σ ∗ ∇ ρ | ) ≤ µ ( ρ | σ − Z | ) < ∞ . This and (2.6) imply ρ ∧ N ∈ H , σ ( µ ) for any N ∈ (0 , ∞ ). By the Poincar´e inequality (4.1) we obtain µ (( ρ ∧ N ) ) ≤ Cµ ( | σ ∗ ∇ ( ρ ∧ N ) | ) + µ ( ρ ) ≤ Cµ ( | σ ∗ ∇ ρ | ) + 1 < ∞ , N ∈ (0 , ∞ ) . By letting N → ∞ we prove ρ ∈ H , σ ( µ ) and (4.2), so that (4.3) follows from (4.4).19elow we will often use the following version of Young’s inequality on a probability space( E, B , ν ) (see [3, Lemma 2.4]):(4.5) ν ( f g ) ≤ log ν (e f ) + ν ( g log g ) , f, g ≥ , ν ( g ) = 1 . The next lemma ensures the existence of invariant probability measure of P t for bounded σ − Z . Lemma 4.2. Assume (H1) . If σ − Z is bounded then the Markov semigroup P t associatedto the SDE (2.3) has a unique invariant probability measure.Proof. According to (b) in the proof of Lemma 3.1, P t has at most one invariant probabilitymeasure. So, it suffices to prove the existence. Letting µ P t be the distribution at time t of the solution to (2.3) with initial distribution µ , we intend to show that the sequence { n R n µ P t d t } n ≥ is tight, so that the weak limit of a weakly convergent subsequence providesan invariant probability measure of P t . To this end, it suffices to find out a positive compactfunction F on R d such that(4.6) 1 n Z n µ ( P t F ) d t ≤ C, n ≥ C > (H1) implies the hyperboundedness of P t . Precisely, by [21,Theorem 1] (see for instance also [35, Theorem 5.1.4]), we have(4.7) k P t k L q ( µ ) → L q ( t ) ( µ ) ≤ exp h β (cid:16) q − q ( t ) (cid:17)i , t > , q > , q ( t ) := 1 + ( q − tκ . Since µ is a probability measure, there exists a compact function W ≥ µ ( W ) < ∞ . Letting F = √ log W which is again a compact function, we have µ (e V ) < ∞ . We nowprove (4.6) for this function F . To this end, we consider the Feyman-Kac semigroup P Ft f ( x ) := E h f ( X xt )e R t F ( X xs )d s i , t ≥ , x ∈ R d . Since µ (e F ) < ∞ , P Ft is a bounded linear operator from L p ( µ ) to L ( µ ) for every t ≥ p > 1. We first observe that P Ft is bounded on L p ( µ ) for any t ≥ p > 1. Let q = √ p . For any non-negative f ∈ L p ( µ ), by Schwarz’s and Jensen’s inequalities, and that µ is P t -invariant, we have 20 (cid:0) | P Ft f | p (cid:1) = Z R d (cid:16) E h f ( X xt )e R t F ( X xs )d s i(cid:17) p µ (d x ) ≤ Z R d (cid:18)(cid:8) E f q ( X xt ) (cid:9)n E e qq − R t F ( X xs )d s o q − (cid:19) q µ (d x )= Z R d ( P t f q ) q (cid:26) t Z t P s e qtq − F d s (cid:27) q ( q − d µ ≤ (cid:8) µ (( P t f q ) q ( t ) ) (cid:9) qq ( t ) (cid:26) Z R d (cid:18) t Z t P s e qtq − F d s (cid:19) q ( t ) q ( q − q ( t ) − q d µ (cid:27) q ( t ) − qq ( t ) ≤ k P t k qL q ( µ ) → L q ( t ) ( µ ) µ (cid:0) { f q } q (cid:1) max (cid:26)(cid:16) µ (cid:16) e q q ( t ) tq ( t ) − q F (cid:17)(cid:17) q ( t ) − qq ( t ) , (cid:16) µ (cid:16) e qtq − F (cid:17)(cid:17) q ( q − (cid:27) = k P t k qL q ( µ ) → L q ( t ) ( µ ) max (cid:26)(cid:16) µ (cid:16) e q q ( t ) tq ( t ) − q F (cid:17)(cid:17) q ( t ) − qq ( t ) , (cid:16) µ (cid:16) e qtq − F (cid:17)(cid:17) q ( q − (cid:27) µ ( f p ) , t ≥ . By µ (e F ) < ∞ and (4.7), this implies k P Ft k L p ( µ ) < ∞ for any t > , and moreover,lim sup t ↓ k P Ft k L p ( µ ) ≤ . Since F ≥ P Ft ≥ 1, we have lim t ↓ k P Ft k L p ( µ ) = 1 . Inparticular, by taking p = 2 and using the semigroup property, we obtain(4.8) E e R n F ( X µ t ) = µ ( P Fn ≤ k P Fn k L ( µ ) ≤ k P F k nL ( µ ) =: c n < ∞ , n ≥ , where X µ t is the solution to (2.2) with initial distribution µ . Now, define R n = exp (cid:20) √ Z n h ( σ − Z )( X µ s ) , d W s i − Z n | ( σ − Z )( X µ s ) | d s (cid:21) , n ≥ . Since σ − Z is bounded, by Girsanov’s theorem we have µ ( P t F ) = E (cid:8) F ( X µ t ) R n (cid:9) , t ∈ [0 , n ] . Then (4.5) and (4.8) imply1 n Z n µ ( P t F )d t = 1 n Z n E (cid:8) F ( X µ t ) R n (cid:9) d t ≤ n log E e R n F ( X µ t )d t + 1 n E (cid:8) R n log R n } ≤ c + 1 n E (cid:8) R n log R n } . (4.9)Since by Girsanov’s theorem˜ W t := W t − √ Z t ( σ − Z )( X µ s )d s, t ∈ [0 , n ]is a d -dimensional Brownian motion under the probability Q n := R n P , we have E (cid:8) R n log R n } = E Q n log R n = E Q n (cid:18) √ Z n h ( σ − Z )( X µ s ) , d ˜ W s i + 14 Z n | ( σ − Z )( X µ s ) | d s (cid:19) ≤ n k σ − Z k ∞ . Combining this with (4.9), we prove (4.6), and hence finish the proof.21 roof of Theorem . . By Lemma 3.1, (H1) implies that (2.3) has a unique non-explosivesolution and the associated Markov semigroup P t is strong Feller with at most one invariantprobability measure. To apply Lemma 4 . 1, we first consider bounded Z with compactsupport, then pass to the general situation by using an approximation argument.(a) Let Z be bounded with compact support. By Lemma 4.2, P t has a unique invariantprobability measure µ . In particular, L ∗ µ = 0, so that by Lemma 4.1(1) we have µ = ρµ for some ρ ∈ H , σ ( µ ) such that (4.3) holds.Since ρ ∈ H , σ ( µ ), f := log( ρ + δ ) ∈ H , σ ( µ ) for all δ > 0. Taking this f in (4.3) weobtain Z R d | σ ∗ ∇ ρ | ρ + δ d µ ≤ Z R d {| σ − Z | · | σ ∗ ∇ log( ρ + δ ) |} d µ = Z R d {| σ − Z | · | σ ∗ ∇ log( ρ + δ ) |} ρ d µ ≤ (cid:18) Z R d ρ | σ − Z | d µ (cid:19) (cid:18) Z R d ρ | σ ∗ ∇ ρ | ( ρ + δ ) d µ (cid:19) ≤ (cid:18) Z R d ρ | σ − Z | d µ (cid:19) (cid:18) Z R d | σ ∗ ∇ ρ | ρ + δ d µ (cid:19) , δ > . Since µ ( | σ ∗ ∇ ρ | ρ + δ ) < ∞ due to ρ ∈ H , σ ( µ ), this implies Z R d | σ ∗ ∇ ρ | ρ + δ d µ ≤ Z R d ρ | σ − Z | d µ , δ > . By letting δ → Z R d (cid:12)(cid:12) σ ∗ ∇√ ρ (cid:12)(cid:12) d µ ≤ Z R d ρ | σ − Z | d µ < ∞ since σ − Z is bounded and µ ( ρ ) = 1 . So, √ ρ ∈ H , σ ( µ ) by (2.6), and the log-Sobolevinequality (2.7) implies(4.11) µ ( ρ log ρ ) ≤ κ Z R d (cid:12)(cid:12) σ ∗ ∇√ ρ (cid:12)(cid:12) d µ + β. Combining this with (4.10) and the Young inequality (4.5), we obtain µ ( | σ ∗ ∇√ ρ | ) ≤ λ log µ (e λ | σ − Z | ) + 14 λ µ ( ρ log ρ ) ≤ λ log µ (e λ | σ − Z | ) + κ λ µ ( | σ ∗ ∇√ ρ (cid:12)(cid:12) ) + β λ . This and (4.10) imply (2.9). 22imilarly, ρ ∈ H , σ ( µ ) implies f = ( ρ + δ ) − ∈ H , σ ( µ ) for δ > 0, so that by (4.3) wehave Z R d | σ ∗ ∇ ρ | ( ρ + δ ) d µ ≤ Z R d (cid:8) ρ | σ − Z | · | σ ∗ ∇ ( ρ + δ ) − | (cid:9) d µ ≤ (cid:18) Z R d ( ρ | σ − Z | ) ( ρ + δ ) d µ (cid:19) (cid:18) Z R d | σ ∗ ∇ ρ | ( ρ + δ ) d µ (cid:19) ≤ p µ ( | σ − Z | ) (cid:18) Z R d | σ ∗ ∇ ρ | ( ρ + δ ) d µ (cid:19) , δ > . Therefore, (2.10) holds.Finally, by [9] the density function ρ is strictly positive, so that by (2.10) and H , σ ( µ ) = W , σ ( µ ) we have log ρ ∈ H , σ ( µ ) if log ρ ∈ L ( µ ). To prove µ ( | log ρ | ) < ∞ , we use thePoincar´e inequality. As explained above that the defective log-Sobolev inequality impliesthat the spectrum of L is discrete, by the irreducibility of the Dirichlet form we see that L has a spectral gap, equivalently, the Poincar´e inequality µ ( f ) ≤ Cµ ( | σ ∗ ∇ f | ) + µ ( f ) , f ∈ H , σ ( µ )holds for some constant C > . Since ρ is strictly positive, we take ε ∈ (0 , 1) such that µ ( ρ ≤ ε ) ≤ . By (2.10) and µ ( ρ ) = 1, for any δ > ρ + δ ) ∈ H , σ ( µ ).Moreover, by the Poincar´e inequality, (2.10) and | log( ρ + δ ) | ≤ ρ + δ + log ε − for ρ ≥ ε ,there exist constants C , C > µ ( | log( ρ + δ ) | ) ≤ Cµ ( | σ ∗ ∇ log( ρ + δ ) | ) + µ (log( ρ + δ )) ≤ C + 2 µ (log( ρ + δ )1 { ρ ≤ ε } ) + 2 µ (log( ρ + δ )1 { ρ>ε } ) ≤ C + 2 µ ( | log( ρ + δ ) | ) µ ( ρ ≤ ε ) + 2 µ ( ρ + δ + log ε − ) ≤ µ ( | log( ρ + δ ) | ) + C , δ ∈ (0 , . Since µ ( | log( ρ + δ ) | ) < ∞ , this implies µ ( | log ρ | ) = lim δ ↓ µ ( | log( ρ + δ ) | ) ≤ C < ∞ . (b) In general, for any n ≥ Z n ( x ) = 1 {| x | + | Z ( x ) |≤ n } Z ( x ) , L n = L + Z n · ∇ . By (a) and | σ − Z n | ≤ | σ − Z | , L n has an invariant probability measure d µ n = ρ n d µ suchthat µ (cid:0) | σ ∗ ∇√ ρ n | (cid:1) ≤ λ − κ (cid:8) log µ (e λ | σ − Z | ) + β (cid:9) < ∞ ,µ ( | σ ∗ ∇ log ρ n | ) ≤ µ ( | σ − Z | ) < ∞ . {√ ρ n } n ≥ is bounded in H , σ ( µ ). Moreover, the defective log-Sobolev in-equality (2.7) implies the existence of a super Poincar´e inequality, and hence the essentialspectrum of L is empty, see [33, Theorem 2.1 and Corollary 3.3]. So, H , σ ( µ ) is compactlyembedded into L ( µ ), i.e. a bounded set in H , σ ( µ ) is relatively compact in L ( µ ). There-fore, for some subsequence n k → ∞ we have √ ρ n k → √ ρ in L ( µ ) for some nonnegative ρ which satisfies (2.9) and (2.10). In particular, ρ n k → ρ in L ( µ ) so that µ := ρµ is a proba-bility measure. Moreover, by using the Poincar´e inequality as in (a), we prove log ρ ∈ L ( µ )so that log ρ ∈ H , σ ( µ ). It remains to show that L ∗ µ = 0 . Since ( L n k ) ∗ µ n k = 0, for any f ∈ C ∞ ( R d ), there exists a constant C > D such that (cid:12)(cid:12)(cid:12)(cid:12) Z R d Lf d µ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z R d (cid:0) ρLf − ρ n k L n k f (cid:1) d µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z D n | Z − Z n k | ρ + (1 + | Z | ) | ρ n k − ρ | o d µ . (4.12)Since µ (e λ | σ − Z | ) < ∞ , we have | Z n | ≤ | Z | ∈ L qloc (d x ) for any q > 1. Then µ (1 D | Z − Z n | q ) → n → ∞ holds for any q > 1. Moreover, the local Harnack inequality (see[8, Corollary 1.2.11]) implies that { ρ n k , ρ } k ≥ is uniformly bounded on the compact set D .Combining these with µ ( | ρ n k − ρ | ) → 0, we may use the dominated convergence theoremto prove µ ( Lf ) = 0 by taking k → ∞ in (4.12). Therefore, L ∗ µ = 0. Then the proof iscomplete. Proof of Theorem . . By Theorem 2.1, the SDE (2.3) has a unique solution and theassociated semigroup P t is strong Feller having at most one invariant probability measure.So, it suffices to prove that the above constructed probability measure µ is the uniqueinvariant probability measure of L and P t . This can be done according to [29] and [8] asfollows.Let b = Z + a ∇ log ρ and b = Z + Z . Then L = tr( a ∇ )+ b ·∇ , and ˆ L := tr( a ∇ )+ b ·∇ is symmetric in L ( µ ). Obviously, (H1) and (2.8) imply that conditions (1.1 ′ )-(1.3 ′ ) and(1.4) in [29] hold for U = R d ; that is, a ij ∈ W , loc (d x ), a is locally uniformly positive definite,and b ∈ L loc (d x ). Moreover, by the Young inequality (4.5), (2.9), (2.8), (2.11) and (4.11),for small enough r > µ ( k a k + | b − b | ) ≤ µ ( ρ | Z | + k σ k · | σ ∗ ∇ ρ | + ρ k σ k ) ≤ µ ( ρ ( | σ − Z | + 3 k σ k )) + µ ( k σ k · | σ ∗ ∇ ρ | ) ≤ r µ ( ρ log ρ ) + 12 r log µ (e r ( | σ − Z | +3 k σ k ) ) + 2 q µ ( ρ k σ k ) µ ( | σ ∗ ∇√ ρ | ) ≤ r µ ( ρ log ρ ) + 12 r log µ (e r ( | σ − Z | +3 k σ k ) )+ 2 q { ε − µ ( ρ log ρ ) + ε − log µ (e ε k σ k ) } µ ( | σ ∗ ∇√ ρ | ) < ∞ . Therefore, by [29, Theorem 1.5, Proposition 1.9 and Proposition 1.10(a)], ( L, C ∞ ( R d )) has aunique closed extension in L ( µ ) which generates a Markov C -semigroup T µt in L ( µ ) such24hat µ is an invariant probability measure. Then, according to [8, Corollary 1.7.3], µ is theunique invariant probability measure of L .On the other hand, according to [29, Theorem 3.5], there is a standard Markov process { ¯ P x } x ∈ R d ∪{ ∂ } which is continuous and non-explosive for µ -a.e. x , such that the associatedsemigroup ¯ P t satisfies Z ∞ e − λt ¯ P t f d t = Z t e − λt T µt f d t, µ -a.e.holds for any f ∈ B b ( R d ) and λ > 0. So, for any f ∈ B b ( R d ) , ¯ P t f = T µt f holds d t × µ -a.e.By the continuity of the process and the strong continuity of T µt in L ( µ ), ¯ P t f = T µt f µ -a.e.for any t ≥ f ∈ C b ( R d ), and hence also for f ∈ L ( µ ) since C b ( R d ) is dense in L ( µ ).That is, ¯ P t is a µ -version of T µt . In particular, µ is ¯ P t -invariant and the probability measure¯ P µ := Z R d P x µ (d x ) on ¯Ω := C ([0 , ∞ ) → R d )solves the martingale problem of ( L, C ∞ ( R d )), so that under this probability space thecoordinate process ¯ X t (¯ ω ) := ¯ ω t for t ≥ ω ∈ ¯Ω is a weak solution to (2.3) with initialdistribution µ (c.f. [24, Proposition 2.1] or [30, § µ ( P t f ) = µ ( ¯ P t f ) for t ≥ f ∈ B b ( R d ). Therefore, µ is an invariant probabilitymeasure of P t . Proof of Theorem 2.4. Obviously, the proof of Theorem 2.3(2) also works if we replace (H1) by (H ′ ) . So, we only need to prove assertion (1). Next, by repeating (b) in the proof ofTheorem 2.3(1), we may and do assume that Z is bounded having compact support, andonly prove that L has an invariant probability measure d µ = ρ d µ with ρ satisfying therequired estimates (2.13) and (2.14). Here, the only thing we need to clarify is that in theright hand side of (4.12) the term (1 + | Z | ) should be replaced by (1 + | Z | + |∇ σ | ) since ∇ σ is no longer locally bounded. This does not make any trouble since |∇ σ | ∈ L loc (d x ) by (H ′ ) , and ( ρ n k − ρ )1 D is uniformly bounded according to [8, Corollarty 1.2.11].Now, we assume that Z is bounded with compact support. Let ˜ V ∈ C ∞ ( R d ) with k ˜ V − V k ∞ ≤ 1, and let ˜ P t be the Markov semigroup generated by ∆ − ∇ ˜ V . Then H , ( µ ) = H , (e − ˜ V ( x ) d x ) , so that (H ′ ) together with the smoothness and positivity-preserving of ˜ P t implies ˜ a n := ˜ P n a ∈ C ( R d → R d ⊗ R d ) , ˜ a n ≥ αI, and (˜ a n ) ij → a ij in H , ( µ ) ∩ L p ( µ ) , ≤ i, j ≤ d. (4.13)Let ˜ L n be defined as L for ˜ a n in place of a ; that is,˜ L n = tr(˜ a n ∇ ) + d X i,j =1 (cid:8) Z i + ∂ j (˜ a n ) ij − (˜ a n ) ij ∂ j V (cid:9) e i . By Lemmas 4.1 and 4.2, ˜ L n has an invariant probability measure ˜ µ n (d x ) := ˜ ρ n (d x ) µ (d x )with ˜ ρ n ∈ H , ( µ ) such that µ ( ˜ ρ n + |∇ ˜ ρ n | ) ≤ Cµ ( ˜ ρ n | Z | ) < ∞ . { ˜ ρ n } n ≥ is uniformly bounded on the compact set D :=supp Z , so this implies that { ˜ ρ n } n ≥ is bounded in H , ( µ ), and hence ˜ ρ n k → ρ in L ( µ )for some subsequence n k → ∞ and some ρ ∈ H , ( µ ). In particular, µ (d x ) := ρ ( x )d x is aprobability measure. We intend to prove L ∗ µ = 0.For any f ∈ C ∞ ( R d ) there exists a constant C ( f ) > | ˜ L n f − Lf | ≤ C ( f ) (cid:0) k∇ ˜ a n − ∇ a k + |∇ V | · k ˜ a n − a k (cid:1) , | ˜ L n f | ≤ C ( f )( k∇ ˜ a n k + k a k · |∇ V | ) . By (4.13), |∇ V | ∈ L pp − ( µ ) included in (H ′ ) , ˜ ρ n k → ρ in L ( µ ), and ˜ L ∗ n ˜ µ n = 0, we areable to use the dominated convergence theorem to derive | µ ( Lf ) | = lim k →∞ | µ ( Lf ) − ˜ µ n k ( ˜ L n k f ) | ≤ lim sup k →∞ µ ( | Lf − ˜ L n k f | ρ + | ˜ L n k f | · | ˜ ρ n k − ρ | ) = 0 . So, L ∗ µ = 0.Since (5.3) and ˜ a n ≥ αI imply (2.7) for (cid:0) √ ˜ a n , κ ′ α (cid:1) in place of ( σ, κ ), by Theorem 2.3 wehave αµ (cid:16) |∇ p ˜ ρ n k | (cid:17) ≤ µ (cid:16)(cid:12)(cid:12)(cid:12)p ˜ a n k ∇ p ˜ ρ n k (cid:12)(cid:12)(cid:12) (cid:17) ≤ αλ − κ ′ α (cid:8) log µ (cid:0) e αλ | (˜ a nk ) − / Z | (cid:1) + β (cid:9) ≤ α α λ − κ ′ (cid:8) log µ (cid:0) e λ | Z | (cid:1) + β (cid:9) ,αµ ( |∇ log ˜ ρ n k | ) ≤ µ (cid:0)(cid:12)(cid:12) (˜ a n k ) − / ∇ log ˜ ρ n k (cid:12)(cid:12) (cid:1) ≤ α µ ( | Z | ) . By using ρ n k + δ to replace ρ n k , and letting first k → ∞ then δ ↓ 0, we prove (2.13) and(2.14) from these two inequalities respectively. The following Sobolev embedding theorem is crucial in the proof. This result can be deducedfrom existing ones, for instance, [26, Corollary 1.4] in the framework of generalized Mehlersemigroup. We include below a brief proof by using the dimension-free Harnack inequalityfor the O-U semigroup. Lemma 5.1. Let (2.15) hold. Then H , ( µ ) is compactly embedded into L ( µ ) ; i.e. boundedsets in H , ( µ ) are relatively compact in L ( µ ) . Proof. Consider the linear SPDE(5.1) d X t = − AX t d t + √ W t , By (2.15), for any initial point x this equation has a unique mild solution X xt = e − At x + √ Z t e − A ( t − s ) d W s , t ≥ , P t f ( x ) := E f ( X xt ) , t ≥ , f ∈ B b ( H ) , x ∈ H is symmetric in L ( µ ) with Dirichlet form E ( f, g ) := µ ( h∇ f, ∇ g i ) , f, g ∈ H , ( µ ) , see for instance [17]. So, by the spectral theory, H , ( µ ) is compactly embedded into L ( µ )if and only if P t is compact for some (equivalently, all) t > 0, both are equivalent to theabsence of the essential spectrum of the generator. By [38, Theorem 3.2.1] with b = 0 and σ = √ K = 0 and λ = , P t satisfies the Harnack inequality(5.2) ( P t f ( x )) ≤ ( P t f ( y )) e | x − y | /t , t > , x, y ∈ H , f ∈ B b ( H ) , which implies that P t has a density with respect to the invariant probability measure µ (see [38, Theorem 1.4.1]). Next, it is well known that the Gaussian measure µ satisfies thelog-Sobolev inequality (see for instance [21])(5.3) µ ( f log f ) ≤ λ µ ( |∇ f | ) , f ∈ H , ( µ ) , µ ( f ) = 1 . This, together with the existence of density of P t with respect to µ for any t > 0, impliesthat P t is compact in L ( µ ) for all t > 0, see [20, Theorem 1.2], [34, Theorems 1.1 and 3.1]or [37, Theorem 1.6.1]. Proof of Theorem . . For any n ≥ 1, let H h n i = { x ∈ H : h x, e i i = 0 , ≤ i ≤ n } be the orthogonal complement of H n := span { e , · · · , e n } . Let π n : H → H n and π h n i : H → H h n i be orthogonal projections. For convenience, besides the orthogonal decomposition H = H n ⊕ H h n i we may regard H as the product space H = H n × H h n i , so that µ = µ ( n )0 × µ h n i for µ ( n )0 = µ ◦ π − n and µ h n i = µ ◦ π − h n i being the marginal distributions of µ on H n and H h n i respectively. Let(5.4) a n ( x ) = π n a ( x ) , Z n ( x ) = π n Z H h n i Z ( x, y ) µ h n i (d y ) , x ∈ H n . By (H2) we have(5.5) h a n v, v i ≥ α | v | , v ∈ H n , and due to Jensen’s inequality,(5.6) µ ( n )0 (e λ | Z n | ) ≤ Z H n e λ R H h n i | Z ( x,y ) | µ h n i (d y ) µ ( n )0 (d x ) ≤ Z H e λ | Z | d µ < ∞ , n ≥ . Let V n ( x ) = P ni =1 λ i x i and L ( n ) = L ( n )0 + Z n · ∇ on H n , where L ( n )0 = n X i,j =1 (cid:16) a ij ∂ i ∂ j + (cid:8) ∂ j a ij − a ij ∂ j V n (cid:9) ∂ i (cid:17) . (H2) imply(5.7) µ ( f log f ) ≤ λ α µ ( h a ∇ f, ∇ f i ) , f ∈ H , ( µ ) , µ ( f ) = 1 , and that (5.5) implies α | a − / n Z n | ≤ | Z n | , we may apply Theorem 2.3(1) to L n on R n ≡ H n for κ = λ α , β = 0 and λα in place of λ , to conclude that L ( n ) has an invariant probabilitymeasure µ n with density function ρ n := d µ ( n ) d µ ( n )0 satisfying √ ρ n ∈ H , ( µ ( n )0 ) and µ ( n )0 (cid:0)(cid:12)(cid:12) ∇√ ρ n (cid:12)(cid:12) (cid:1) ≤ α µ ( n )0 ( |√ a n ∇√ ρ n | ) ≤ λ α λ λ − µ ( n )0 (e λα | a − / n Z n | ) ≤ λ α λ λ − µ ( n )0 (e λ | Z n | ) ≤ λ α λ λ − µ (e λ | Z | ) < ∞ , n ≥ , (5.8)where the last step is due to Jensen’s inequality and the definitions of Z n and µ ( n )0 . Moreover, µ ( n )0 ( |∇ log ρ n | ) ≤ α µ ( n )0 ( |√ a n ∇ log ρ n | ) ≤ µ ( n )0 ( | a − / n Z n | ) α ≤ µ ( n )0 ( | Z n | ) α ≤ µ ( | Z | ) α < ∞ , n ≥ . (5.9)Letting ¯ ρ n = ρ n ◦ π n , (5.8) implies that {√ ¯ ρ n } n ≥ is bounded in H , ( µ ). By Lemma 5.1,there exists a subsequence n k → ∞ and some positive ρ ∈ L ( µ ) with √ ρ ∈ H , ( µ ) suchthat √ ¯ ρ n k → √ ρ in L ( µ ), (2.20) and (2.21) hold. Then log ρ ∈ H , ( µ ) as shown in theend of the proof of Theorem 2.3(1) using the Poincar´e inequality. In particular, µ := ρµ isa probability measure on H . It remains to show that L ∗ µ = 0.By the definition of Z n , we have ¯ Z n := Z n ◦ π n = π n µ ( Z | π n ) , where µ ( ·| π n ) is theconditional expectation of µ given π n . Since µ ( | Z | ) < ∞ , by the martingale convergestheorem, µ ( Z | π n ) → Z in L ( µ ), and hence, ¯ Z n → Z in L ( µ ) as well. By the continuityof a , ¯ a n := a n ◦ π n → a pointwise. Noting that for any f ∈ F C ∞ there exist l ∈ N and aconstant C ( f ) > | µ ( Lf ) | = | µ ( Lf ) − µ n k ( L n k f ) |≤ C ( f ) µ (cid:16) ρ {| Z − ¯ Z n k | + l X i,j =1 | ( a − ¯ a n k ) ij |} (cid:17) + C ( f ) µ (cid:16)n | ¯ Z n k | + l X i,j =1 | (¯ a n k ) ij | o | ρ − ¯ ρ n k | (cid:17) holds for n k ≥ l , to prove µ ( Lf ) = 0 by using the dominated convergence theorem, it sufficesto verify the uniform integrability of { ¯ ρ n ( | ¯ Z n | + | a ij ◦ π n | ) } n ≥ in L ( µ ) for every i, j ≥ ε ∈ (0 , 1) there exists a constant C ( ε ) > | ¯ Z n | + | a ij ◦ π n | ) ¯ ρ n ≤ e ε | ¯ Z n | ε + e ε | a ij ◦ π n | ε + C ¯ ρ n { log(e + ¯ ρ n ) } ε , n ≥ . Since µ ( n )0 ( f ) = µ ( f ◦ π n ) for f ∈ L ( µ ( n )0 ), this implies the desired the uniform integrabilityby (2.17), (5.6), (5.8) and µ ( ¯ ρ n log ¯ ρ n ) ≤ λ µ ( |∇√ ¯ ρ n | ) = 2 λ µ ( |∇√ ρ n | )due to the log-Sobolev inequality (5.3). 28 roof of Theorem . . The desired assertion can be deduced from [29]. Since a is boundedand (H2) holds, we have H , ( µ ) = H , σ ( µ ) . Let µ be a probability measure µ on H suchthat the form E µ ( f, g ) := µ ( h a ∇ f, ∇ g i ) , f, g ∈ F C ∞ is closable in L ( µ ), and let ( L µ , D ( L µ )) be the generator of the closure ( E µ , H , ( µ )). More-over, let β ∈ L ( H → H ; µ ) such that(5.10) µ ( h β, ∇ f i ) = 0 , f ∈ H , ( µ ) . Then, according to Proposition 1.3, Theorem 1.9 and Proposition 1.10 in [29, Part II], wehave the following assertions for L := L µ + β · ∇ :(i) ( L, F C ∞ b ) is dissipative and hence closable in L ( µ ), whose closure ( ¯ L, D ( ¯ L )) generatesa Markovian C -semigroup of contraction operators ( T t ) t ≥ on L ( µ ), D ( ¯ L ) ⊂ H , ( µ ),and(5.11) µ ( h∇ f, β − a ∇ g i ) = µ ( g ¯ Lf ) , f ∈ D ( ¯ L ) ∩ B b ( H ) , g ∈ H , ( µ ) ∩ B b ( H ) . (ii) There exists a standard continuous Markov process { ¯ P x } x ∈ H whose semigroup ¯ P t sat-isfies(5.12) Z ∞ e − λt ¯ P t f d t = Z ∞ T t f d t, µ -a.e. , λ > , f ∈ B b ( H ) . As shown in the proof of Theorem 2.3(2), (5.12) implies that ¯ P t is a µ -version of T t .Now, let L = L + Z · ∇ and µ = ρµ be in Theorem 2.5. We intend to verify the aboveconditions such that assertions (i) and (ii) hold.Firstly, √ ρ ∈ H , ( µ ) implies ∇ log ρ ∈ L ( µ ) and µ ( |∇ ρ | ) ≤ q µ ( |∇√ ρ | ) µ ( ρ ) < ∞ . Consider the operator L µ := L + a ∇ log ρ, f ∈ F C ∞ . By the symmetry of L in L ( µ ), the boundedness of a , ∇ log ρ ∈ L ( µ ) , ∇ ρ ∈ L ( µ ) andnoting that H , ( µ ) is dense in H , ( µ ), we obtain µ ( f L µ g ) = µ ( f h∇ log ρ, a ∇ g i ) + µ ( f ρL g )= µ ( f h∇ log ρ, a ∇ g i ) − µ ( ∇ ( f ρ ) , a ∇ g i ) = − µ ( h∇ f, a ∇ g i ) , f, g ∈ F C ∞ . Thus, the form ( E µ , F C ∞ ) is closable in L ( µ ) with generator extending ( L µ , F C ∞ ).Next, let β = Z − a ∇ log ρ. We have L = L µ + β · ∇ on F C ∞ . Since L ∗ µ = 0 and µ ( h∇ ρ, ∇ f i ) = − µ ( ρL f ) for f ∈ F C ∞ , we have µ ( h β, ∇ f i ) = µ ( h ρZ − a ∇ ρ, ∇ f i )= µ ( h Z, ∇ f i ) + µ ( ρL f ) = µ ( Lf ) = 0 , f ∈ F C ∞ . (5.13) 29oting that (2.20) and the boundedness of a imply µ ( | a ∇ log ρ | ) ≤ k a k µ ( |∇√ ρ | ) < ∞ , while by the Young inequality (4.5) and the log-Sobolev inequality (5.3) µ ( | Z | ) = µ ( ρ | Z | ) ≤ λ log µ (e λ | Z | ) + 1 λ µ ( ρ log ρ ) ≤ λ log µ (e λ | Z | ) + 2 λ λ µ ( |∇√ ρ | ) + 1 λ < ∞ , we have µ ( | β | ) < ∞ for β := Z − a ∇ log ρ . So, (5.10) follows from (5.13).In conclusion, the above assertions (i) and (ii) hold for the present situation. Combining(5.10) with (5.11) for g = 1 and T t f in place of f , we obtaindd t µ ( T t f ) = µ ( LT t f ) = µ ( h∇ T t f, β i ) = 0 , f ∈ F C ∞ , t ≥ . Therefore, µ is an invariant probability measure of T t , and the proof is finished since ¯ P t is a µ -version of T t . Proof of Theorem . . Since V n ( x ) := P ni =1 λ i x i on H n satisfies |∇ V n | ∈ L ( µ ( n )0 ) for all q > (H ′ ) and (5.7) imply that (H ′ ) holds for ( a n , V n , µ ( n )0 ) in place of ( a, V, µ ) with κ ′ = αλ and β = 0 . So, by repeating the proof of Theorem 2.5 using Theorem 2.4 in placeof Theorem 2.3(1), we prove the desired assertions. We first prove the non-explosion of the weak solution constructed from the Girsnaov trans-form of the linear SPDE (5.1), then prove the strong Feller property of the associated Markovsemigroup. The Feller property, together with the pathwise uniqueness for µ -a.e. startingpoints due to [15], implies that the constructed Markov process is the unique Feller processsolving (2.3) weakly. Noting that in the present case we have d = ∞ , the estimate (3.4)derived in the finite-dimensional case does not make sense. To construct the desired weaksolution we need to establish a reasonable infinite-dimensional version of (3.4). We will soonfind out that this is non-trivial at all. If we start from the Harnack inequality (5.2), it isstandard that ( P t f ( x )) p ≤ µ ( f p ) µ (e −| x −·| p /t ) ≈ e c ( x ) /t for some constant c ( x ) > t > 0. The hard point is that R t e c ( x ) / ( ps ) d s = ∞ forany t > p > 1, so that the argument we used in the finite-dimensional case is invalid.To kill this high singularity for small time t , we will use a refined version of the Harnackinequality and make a clever choice of reference measure ν t on [0 , t ] to replace the Lebesguemeasure. 30 .1 Construction of the weak solution We first construct weak solutions to (2.3) using the Girsanov transform. For any x ∈ H , let X xt solve (5.1) with X = x . Let(6.1) R xs,t := exp (cid:20) √ Z ts h Z ( X xr ) , d W r i − Z ts | Z ( X xr ) | d r (cid:21) , t ≥ s ≥ . By Girsanov’s theorem, if ( R xt ) t ≥ := ( R x ,t ) t ≥ is a martingale, then for any T > f W xt := W t − √ Z t Z ( X xs )d s, t ∈ [0 , T ]is a cylindrical Brownian motion under the weighted probability Q xT := R xT P , so that( X xt , f W xt ) t ∈ [0 ,T ] is a weak solution to (2.22) starting at x . To prove that ( R xt ) t ≥ is a martin-gale, it suffices to verify the Novikov condition(6.2) E e R t | Z ( X xs ) | d s < ∞ , x ∈ H for some t > . Indeed, by the Markov property, this condition implies that ( R xs,t ) t ∈ [ s,s + t ] is a martingale for all x ∈ H and s ≥ 0, and thus ( R xt ) t ≥ is a martingale for all x ∈ H byinduction: if ( R xt ) t ∈ [0 ,nt ] is a martingale for some n ≥ 1, then for any nt ≤ s < t ≤ ( n + 1) t we have E ( R xt | F s ) = R xs E ( R xs,t | F s ) = R xs . Therefore, the condition (6.2) implies that ( X xt , f W xt ) t ∈ [0 ,T ] is a weak solution to (2.22) forany T > x ∈ H . Let P t ( x, d y ) be the distribution of X xt under Q xt , and let(6.3) P t f ( x ) = E Q xt f ( X xt ) = E (cid:8) f ( X xt ) R xt (cid:9) , f ∈ B b ( H ) , t ≥ , x ∈ H . By the Markov property of X t under P , it is easy to see that P t is a Markov semigroup on B b ( H ), i.e. { P t ( x, d y ) : t ≥ , x ∈ H } is a Markov transition kernel.To verify condition (6.2), we introduce a refined version of the Harnack inequality (5.2).For each i ≥ P ,it be the diffusion semigroup generated by L ,i f := f ′′ − λ i f ′ on R . By[32, Lemma 2.1] for K = − λ i and g ( s ) = e − Ks , we have( P ,it f ( x )) p ≤ ( P ,it f p ( y )) exp (cid:20) pλ i | x − y | p − λ i t − (cid:21) , t > , p > , f ∈ B + ( R ) , x, y ∈ R . By regarding P ,it as a linear operator on B b ( H ) acting on the i -th component x i := h x, e i i ,we have P t = Q ∞ i =1 P ,it , so that this Harnack inequality leads to( P t f ( x )) p ≤ P t f p ( y ) exp (cid:20) p p − ∞ X i =1 λ i | x i − y i | e λ i t − (cid:21) , t > , f ∈ B + b ( H ) , x, y ∈ H p > 1. Noting that µ is an invariant probability measure of P t , by taking p = 2 weobtain(6.4) ( P t f ( x )) Z H exp (cid:20) − ∞ X i =1 λ i ( x i − y i ) e λ i t − (cid:21) µ (d y ) ≤ µ ( f ) , x ∈ H , t > , f ∈ L ( µ ) . Observing that λ i ( x i − y i ) e λ i t − λ i y i λ i (e λ i t + 1)2(e λ i t − (cid:16) y i − x i e λ i t + 1 (cid:17) + λ i x i e λ i t + 1 , by (2.16) we have Z H exp (cid:20) − ∞ X i =1 λ i ( x i − y i ) e λ i t − (cid:21) µ (d y )= ∞ Y i =1 √ λ i √ π Z R exp (cid:20) − λ i ( x i − y i ) e λ i t − − λ i y i (cid:21) d y i = exp (cid:20) − ∞ X i =1 λ i x i e λ i t + 1 (cid:21)(cid:18) ∞ Y i =1 e λ i t − λ i t + 1 (cid:19) , t > , x ∈ H . So, (6.4) reduces to(6.5) P t f ( x ) ≤ p µ ( f ) Γ x ( t ) , x ∈ H , t > , f ∈ L ( µ ) , where due to (2.15),Γ x ( t ) := exp (cid:20) ∞ X i =1 λ i x i e λ i t + 1 (cid:21)(cid:18) ∞ Y i =1 e λ i t + 1e λ i t − (cid:19) ≤ exp (cid:20) ∞ X i =1 λ i x i e λ i t + 1 (cid:21)(cid:18) ∞ Y i =1 (cid:16) λ i t (cid:17)(cid:19) < ∞ , t > , x ∈ H . (6.6)Moreover, using the stronger condition P ∞ i =1 λ − θi < ∞ for some θ ∈ (0 , 1) included in (H3) ,and noting that log(1 + r ) ≤ cr θ for some constant c > r ≥ 0, we obtainΨ( t, x ) := Z t log Γ x ( s )d s = 14 ∞ X i =1 Z t n λ i x i e λ i s + 1 + log (cid:16) λ i s (cid:17) d s o d s ≤ ∞ X i =1 (cid:26) Z t λ i x i e − λ i s d s + cλ θi Z t r − θ d r (cid:27) ≤ ∞ X i =1 x i (1 − e − λ i t ) + Ct − θ < ∞ , t > , x ∈ H (6.7) 32or some constant C > . For later use we deduce from this that(6.8) lim sup t → sup y → x Ψ( t, y ) ≤ 12 lim sup t → sup y → x (cid:26) ∞ X i =1 x i (1 − e − λ i t ) + | x − y | + Ct − θ (cid:27) = 0 . Since (6.6) implies Γ x ( s ) ∈ (1 , ∞ ), for every t > β x ( t ) := Z t d s Γ x ( s ) ∈ (0 , t ] , so that ν t,x (d s ) := 1 [0 ,t ] ( s ) β x ( t )Γ x ( s ) d s is a probability measure on [0 , t ] . Noting that β x ( t ) t R t Γ x ( s ) ν t,x (d s ) = 1 and log (cid:0) β x ( t ) t Γ x ( s ) (cid:1) ≤ log Γ x ( s ), the Young inequality (4.5) yields Z t | Z ( X xs ) | d s = 2 tλ Z t (cid:16) λ | Z ( X xs ) | (cid:17)(cid:16) β x ( t ) t Γ x ( s ) (cid:17) ν t,x (d s ) ≤ tλ log ν t,x (cid:0) e λ | Z ( X x · ) | (cid:1) + 2 tλ Z t n β x ( t ) t Γ x ( s ) log (cid:16) β x ( t ) t Γ x ( s ) (cid:17)o ν t,x (d s ) ≤ tλ log ν t,x (cid:0) e λ | Z ( X x · ) | (cid:1) + 2 λ Ψ( t, x ) , t ≥ , x ∈ H . Combining this with (6.5) for f = e λ | Z | , (6.7) and µ (e λ | Z | ) < ∞ , we arrive at E exp (cid:20) γ Z t | Z ( X xs ) | d s (cid:21) ≤ e γλ Ψ( t,x ) E (cid:26) Z t e λ | Z ( X xs ) | ν t,x (d s ) (cid:27) γtλ ≤ e γλ Ψ( t,x ) (cid:26) Z t (cid:8) P s e λ | Z | ( x ) (cid:9) ν t,x (d s ) (cid:27) γtλ ≤ e γλ Ψ( t,x ) (cid:26) Z t q µ (e λ | Z | )Γ x ( s ) ν t,x (d s ) (cid:27) γtλ = e γλ Ψ( t,x ) (cid:16) tβ x ( t ) q µ (e λ | Z | ) (cid:17) γtλ =: Λ( t, x, γ ) < ∞ , x ∈ H , γ > , t ∈ (cid:16) , λ γ i . (6.9)By taking γ = , we prove (6.2) for t = 2 λ . P t By the Harnack inequality (5.2), P t is strong Feller having strictly positive density withrespect to µ (see [40, Proposition 3.1(1)]). Then as in (b) and (c) in the proof of Lemma3.1, we may prove the same property for P t using (6.3) and (6.9). To save space, we onlyprove here the strong Feller property. 33or any t > , by the semigroup group property of P t , (6.3), and the strong Feller propertyof P t , we obtainlim sup y → x | P t f ( x ) − P t f ( y ) | = lim sup r → lim sup y → x | P r ( P t − r f )( x ) − P r ( P t − r f )( y ) |≤ lim sup r → lim sup y → x n | P r ( P t − r f )( x ) − P r ( P t − r f )( x ) | + (cid:12)(cid:12) E (cid:2) ( P t − r f )( X xr )( R xr − − ( P t − r f )( X yr )( R rr − (cid:3)(cid:12)(cid:12)o ≤ k f k ∞ lim sup r → lim sup y → x E (cid:0) | R xr − | + | R yr − | (cid:1) . (6.10)Recalling that R yr = R y ,r , by (6.1) we have E | R yr − | = E ( R yr ) − ≤ (cid:16) E e R r | Z ( X ys ) | d s (cid:17) − , y ∈ R d . So, according to (6.10), P t is strong Feller provided(6.11) lim sup r → lim sup y → x E exp (cid:20) Z r | Z ( X ys ) | d s (cid:21) = 1 . Recall that β x ( t ) = R t s Γ x ( s ) d s. By Jensen’s inequality and (6.7) we havelog β x ( t ) t = − log (cid:18) t Z t d s Γ x ( s ) (cid:19) ≤ − t Z t n log 1Γ x ( s ) o d s = Ψ( t, x ) t . Combining this with (6.8) and (6.9), we obtainlim r → lim sup y → x Λ( r, y, ≤ lim r → lim sup y → x e λ Ψ( r,y ) (cid:16) e r Ψ( r,y ) q µ (e λ | Z | ) (cid:17) rλ = lim r → lim sup y → x e λ Ψ( r,y ) = 1 . Combining this with (6.9), we prove (6.11). To prove that P t is the unique Feller Markov semigroup associated to (2.22), we recall thepathwise uniqueness for µ -a.e. initial points. By [15, Theorem 1], there exists an µ -null set H such that for any x / ∈ H , the SPDE (2.22) has at most one mild solution starting at x upto life time. Combining this with the weak solution constructed in (a), we see that for anyinitial point x / ∈ H , the SPDE (2.22) has a unique mild solution X xt which is non-explosivewith distribution P t ( x, d y ). So, if there exists another Feller transition probability kernel¯ P t ( x, d y ) associated to (2.22), then ¯ P t ( x, d y ) = P t ( x, d y ) for x / ∈ H . Since H \ H is dense34n H , by the Feller property these transition probability kernels are weak continuous in x ,so that ¯ P t ( x, d y ) = P t ( x, d y ) for all x ∈ H .Next, according to [40, Proposition 3.1(3)], to show that P t has at most one invariantprobability measure, it suffices to prove for instance the Harnack inequality(6.12) ( P t f ) ( x ) ≤ ( P t f )( y ) H t ( x, y ) , x, y ∈ H , f ∈ B b ( H )for some t > H t : H → (0 , ∞ ) . By (6.3) and (5.2), we have( P t f ( x )) = (cid:8) E [ f ( X xt ) R xt ] (cid:9) ≤ (cid:8) P t f ( x ) E ( R xt ) (cid:9) ≤ (cid:8) ( P t f )( y ) (cid:9) E ( R xt ) = (cid:8) E f ( X yt ) (cid:9) [ E ( R xt ) ] ≤ (cid:8) E [ f ( X yt ) R yt ] (cid:9) · (cid:8) E ( R yt ) − (cid:9) E ( R xt ) = { P t f ( y ) } · (cid:8) E ( R yt ) − (cid:9) [ E ( R xt ) ] . By (6.9) and the definition of R · t , it is easy to see that when t > { E ( R yt ) − } [ E ( R xt ) ] ≤ H t ( x, y ) holds for some measurable function H t : H → (0 , ∞ ) . There-fore, (6.12) holds. P t -invariance of µ and estimates on the density Finally, we prove that µ in Theorem 2.5 is an invariant probability measure of P t . Let µ and ¯ P x be in Theorem 2.5, according to the proof of Theorem 2.3(2) we conclude that¯ P µ := R H ¯ P x µ (d x ) is the distribution of a weak solution to (2.22) with initial distribution µ .Since µ is absolutely continuous with respect to µ , the uniqueness for µ -a.e. initial pointsimplies that the weak solution starting from µ is unique, so that µ ( P t f ) = µ ( ¯ P t f ) for t ≥ f ∈ B b ( H ) . Since µ is ¯ P t -invariant, it is P t -invariant as well. Since Theorem 2.5 implies √ ρ ∈ H , ( µ ), (2.20) and (2.21), it remains to prove log ρ ∈ H , ( µ ).By µ ( ρ ) = 1 and √ ρ ∈ H , ( µ ), we have log( ρ + δ ) ∈ H , ( µ ) for all δ > 0. Combiningthis with (2.21) we conclude that log ρ ∈ H , σ ( µ ) provided µ ( ρ > 0) = 1 with µ ( | log ρ | ) < ∞ . It is well known that the Gaussian measure µ satisfies the Poincar´e inequality µ ( f ) ≤ λ µ ( |∇ f | ) + µ ( f ) , f ∈ H , σ ( µ ) . Then, as shown in the last step in the proof of Theorem 2.3(1), µ ( | log ρ | ) < ∞ followsfrom (2.21) if µ ( ρ > 0) = 1. Thus, we only need to prove µ ( ρ > 0) = 1.Recalling that R xt = R x ,t for R xs,t defined in (6.1), by (6.3) and (6.9) we may find out aconstant t > H ∈ C ( H → (0 , ∞ )) such that for any f ∈ B + b ( H ), (cid:0) P t f ( x ) (cid:1) = (cid:0) E f ( X xt ) (cid:1) ≤ (cid:0) E [ f ( X xt ) R xt ] (cid:1) E (cid:2) ( R xt ) − (cid:3) = ( P t f ( x )) E (cid:2) ( R xt ) − (cid:3) ≤ H ( x ) P t f ( x ) , x ∈ H . Then for any measurable set A ⊂ H with µ ( A ) > 0, we have(6.13) µ ( A ) = µ ( P t A ) ≥ µ (cid:18) ( P t A ) H (cid:19) . 35n the other hand, by µ ( P t A ) = µ ( A ) > 0, there exists y ∈ H such that P t A ( y ) > P t A ( x ) ≥ ( P t A ( y )) e − C | x − y | t > , x ∈ H . Combining this with (6.13) and H > 0. Therefore, µ is absolutely continuous with respectto µ and hence, µ ( ρ > 0) = 1 . Let M be a d -dimensional differential manifold without boundary which is equipped with a(not necessarily complete) C -metric such that the curvature is well defined and continuous.Let ∆ and ∇ be the corresponding Laplace-Beltrami operator and the gradient operatorrespectively. Then for any V ∈ C ( M ), the operator L := ∆ + ∇ V generates a uniquediffusion process up to life time. Let ( X xt ) t ∈ [0 ,ζ ( x )] be the diffusion process starting at x withlife time ζ ( x ). Then the associated Dirichlet semigroup is given by P t f ( x ) := E (cid:8) { t<ζ ( x ) } f ( X xt ) (cid:9) , x ∈ M, t ≥ , f ∈ B b ( M ) . For any f ∈ B + b ( M ) := { f ∈ B b ( M ) : f ≥ } , define E P t ( f ) = P t ( f log f ) − ( P t f ) log P t f, t ≥ . Let ρ be the Riemannian distance. By the locally compact of the manifold we may take R ∈ C ( M → (0 , ∞ )) such that B ρ ( x, R ( x )) := (cid:8) y ∈ M : ρ ( x, y ) ≤ R ( x ) (cid:9) is compact for all x ∈ M . When the metric is complete this is true for all R ∈ C ( M → (0 , ∞ ).We will use this function R to establish the local Harnack inequality. Theorem 7.1. There exists a function H ∈ C ( M → (0 , ∞ )) such that (7.1) |∇ P t f ( x ) | ≤ δE P t ( f )( x ) + H ( x ) (cid:16) δ + 1 δ ( t ∧ (cid:17) , t > , δ ≥ R ( x ) , f ∈ B + b ( M ) . Consequently, for any p > there exists a function F ∈ C ( M → (0 , ∞ )) such that for any t > and f ∈ B + b ( M ) , (7.2) ( P t f ( x )) p ≤ ( P t f p ( y )) exp (cid:20) F ( x ) ρ ( x, y ) t ∧ F ( x ) (cid:21) , x, y ∈ M with ρ ( x, y ) ≤ F ( x ) . Proof. According to [2], it is easy to prove (7.2) from (7.1). When the metric is complete,an estimate of type (7.1) for all δ > R ( x ) arbitrarily large as in[3]. Below we figure out the proof in the present case.361) To prove (7.1), we fix f ∈ B + b ( M ). By using fP t f ( x ) replace f , we may and do assumethat P t f ( x ) = 1 at a fixed point x so that E P t ( f )( x ) = P t ( f log f )( x ) . Now, let us check the proof of Theorem 1.1 in [3] (pages 3666-3667), where the partbefore (4.5) has nothing to do with the completeness; that is, with the compact set D := B ρ ( x, R ( x )), all estimates therein before (4.5) apply to the present setting. More precisely,letting τ ( x ) = inf { t ≥ X xt / ∈ D } , we have ((4.1) in [3])(7.3) |∇ P t f ( x ) | ≤ I + I , where ((4.2) in [3])(7.4) I ≤ δ E (cid:8) { t<τ ( x ) } ( f log f )( X xt ) (cid:9) + δ e + C ( x ) (cid:16) δt (cid:17) , δ > , t > C ∈ C ( M → (0 , ∞ )) depending only on d and curvature of the operator L ; and moreover ((4.5) in [3]),(7.5) I ≤ δ E (cid:8) { τ ( x ) ≤ t<ζ ( x ) } ( f log f )( X xt ) (cid:9) + δ e + δ log E e R ( x ) δτ ( x ) + A ( x ) , δ > , t > A ( x ) := sup r> (cid:8) C ( x ) √ r log(e + r ) − r (cid:9) , which is finite and continuous in x . Now,due to the restriction of R ( x ), we have to take large enough δ > δ by δ ∧ δ to replace α ∧ E e R ( x ) δτ ( x ) ≤ Z ∞ (9 u + 1)e − u d u =: A ′ < ∞ , δ ≥ R ( x ) . Combining the with (7.3)-(7.5), we prove (7.1) for some H ∈ C ( M → (0 , ∞ )) . (2) Since H, R are strictly positive and continuous, and B ρ ( x, R ( x )) is compact for every x , ¯ H ( x ) := sup B ρ ( x,R ( x )) H and ˆ R ( x ) := inf B ρ ( x,R ( x )) R are strictly positive continuous functions in x . For any p > 1, let G ( x ) = p − p ¯ H ( x ) ∧ ˆ R ( x ) , x ∈ M. Then (7.1) implies |∇ P t f ( y ) | ≤ δE P t ( f )( y ) + ¯ H ( x ) (cid:16) δ (1 ∧ t ) + δ (cid:17) , y ∈ B ρ ( x, G ( x )) , δ ≥ R ( x )37or f ∈ B + b ( M ) . So, letting γ : [0 , → M be the minimal geodesic from x to y with | ˙ γ s | = ρ ( x, y ) for s ∈ [0 , , letting β ( s ) = 1 + s ( p − δ := p − pρ ( x,y ) ≥ R ( x ) , we obtaindd s (cid:8) log P t f β ( s ) (cid:9) pβ ( s ) = p ( p − E P t ( f β ( s ) ) β ( s ) P t f β ( s ) + p h∇ P t f β ( s ) , ˙ γ s i β ( s ) P t f β ( s ) ( γ s ) ≥ pρ ( x, y ) β ( s ) P t f β ( s ) ( γ s ) (cid:26) p − pρ ( x, y ) E P t ( f β ( s ) ) − |∇ P t f β ( s ) | (cid:27) ( γ s ) ≥ − pρ ( x, y ) β ( s ) P t f β ( s ) ( γ s ) (cid:26) ¯ H ( x ) (cid:0) P t f β ( s ) ( γ s ) (cid:1)(cid:16) pρ ( x, y )( p − t ∧ 1) + p − pρ ( x, y ) (cid:17)(cid:27) ≥ − ¯ H ( x ) (cid:16) p ρ ( x, y ) ( p − t ∧ 1) + 1 (cid:17) , s ∈ [0 , , ρ ( x, y ) ≤ G ( x ) . Integrating over [0 , 1] with respect to d s , we prove (7.2) for F := p ¯ Hp − ∨ G . Acknowledgements. The author would like to thank Vladimir Bogachev, Chenggui Yuanand the referees for helpful comments and corrections. References [1] S. Albeverio, V. Bogachev, M. R¨ockner, On uniqueness of invariant measures for finite-dimensional diffusions, Comm. Pure Appl. Math. 52(1999), 325–362.[2] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Harnack inequality and heat kernel estimateson manifolds with curvature unbounded below, Bull. Sci. Math. 130(2006), 223–233.[3] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimates and Harnack inequalitieson non-compact Riemannian manifolds, Stoch. Proc. Appl. 119(2009), 3653–3670.[4] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Equivalent log-Harnack and gradient forpoint-wise curvature lower bound, Bull. Math. Sci. 138(2014), 643–655.[5] D. Bakry, D. and M. Emery, Hypercontractivit´e de semi-groupes de diffusion , C. R.Acad. Sci. Paris. S´er. I Math. 299(1984), 775–778.[6] R. F. Bass, Z. Q. Chen, Brownian motion with singular drift, Ann. Probab. 31(2003),791–817.[7] V. I. Bogachev, N. V. Krylov, M. R¨ockner, On regularity of transition probabilities andinvariant measures of singular diffusions under minimal conditions, Comm. Part. Diff.Equat. 26 (2001), 2037–2080. 388] V.I. Bogachev, N.V. Krylov, M. R¨ockner, Elliptic and parabolic equations for measures, Russ. Math. Surv. 64(2009), 973–1078.[9] V.I. Bogachev, M. R¨ockner, Regularity of invariant measures on finite and infinite di-mensional spaces and applications, J. Funct. Anal. 133(1995), 168–223.[10] V.I. Bogachev, M. R¨ockner, A generalization of Hasminskii’s theorem on existence ofinvariant measures for locally integrable drifts, Theo. Probab. Appl. 45(2002), 363–378.[11] V.I. Bogachev, M. R¨ockner, Elliptic equations for measures on infinite dimensionalspaces and applications, Probab. Theo. Relat. Fields 120(2001), 445–496.[12] V. I. Bogachev, M. R¨ockner, F.-Y. Wang, Elliptic equations for invariant measures onfinite and infinite dimensional manifolds, J. Math. Pure Appl. 80(2001), 177–221.[13] P. Cattiaux, A. Guillin, L. Wu, A note on Talagrand’s transportation inequality andlogarithmic Sobolev inequality, Probab. Theo. Relat. Fields 148(2010), 285–304.[14] P. E. Chaudru de Raynal, Strong existence and uniqueness for stochastic differentialequation with H¨orlder drift and degenerate noise, to appear in Ann. Inst. Henri Poincar´eProbab. Stat. http://arxiv.org/abs/1205.6688.[15] G. Da Prato, F. Flandoli, M. R¨ockner, A. Yu. Veretennikov, Strong uniqueness forstochastic evolution equations with unbounded measurable drift term, J. Theo. Probab.28(2015), 1571–1600.[16] G. Da Prato, M. R¨ockner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theo. Relat. Fields 124(2002), 261–303.[17] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Uni-versity Press, Cambridge, 1992.[18] E. B. Davies, B. Simon, Ultracontractivity and heat kernel for Schr¨odinger operatorsand Dirichlet Laplacians, J. Funct. Anal. 59(1984), 335–395.[19] E. Priola, F.-Y. Wang, Gradient estimates for diffusion semigroups with singular coef-ficients, J. Funct. Anal. 236(2006), 244–264.[20] F. Gong, F.-Y. Wang, Functional inequalities for uniformly integrable semigroups andapplication to essential spectrum, Forum Math. 14(2002), 293–313.[21] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97(1976), 1061–1083.[22] I. Gy¨ongy, T. Martinez, On stochastic differential equations with locally unbounded drift, Czechoslovak Math. J. 51(2001), 763–783.[23] R.Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff and noordhoff,1980. 3924] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (Sec-ond Edition), North-Holland, 1989.[25] N. V. Krylov, M. R¨ockner, Strong solutions of stochastic equations with singular timedependent drift, Probab. Theo. Relat. Fields 131(2005), 154–196.[26] M. R¨ockner, F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric)diffusion semigroups on manifolds, Forum Math. 15(2003), 893–921.[27] M. R¨ockner, F.-Y. Wang, Harnack and functional inequalities for generalized Mehlersemigroups, J. Funct. Anal. 203(2003), 197–234.[28] M. R¨ockner, F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric)diffusion semigroups on manifolds, Forum Math. 15(2003), 893–921.[29] W. Stannat, (Nonsymmetric) Dirichlet operators on L : Existence, uniqueness andassociated Markov processes, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28(1999) 99–140.[30] D. Stroock, S.R.S. Varadhan, Multidimensional Diffusion processes, Springer, 1979.[31] M. Takeda, G. Trutnau, Conservativeness of non-symmetric diffusion processes gener-ated by perturbed divergence forms, Forum Math. 24(2012), 419–444.[32] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theo. Relat. Fields 109(1997), 417–424.[33] F.Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal. 170(2000), 219–245.[34] F.-Y. Wang, Functional inequalities, semigroup properties and spectrum estimates, Inf.Dimens. Anal. Quant. Probab. Relat. Top. 3(2000), 263–295.[35] F.-Y. Wang, Functional Inequalities, Markov Processes and Spectral Theory, SciencePress, 2005.[36] F.-Y. Wang, Logarithmic Sobolev inequalities: conditions and counterexamples, J. Op-erat. Theo. 46(2001), 183–197.[37] F.-Y. Wang, Analysis for Diffusion Processes on Riemannian Manifolds, World Scien-tific, 2014.[38] F.-Y. Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer, 2013.[39] F.-Y. Wang, Gradient estimates and applications for SDEs in Hilbert space with multi-plicative noise and Dini drift, J. Diff. Equat. 260(2016), 2792–2829.[40] F.-Y. Wang, C. Yuan, Harnack inequalities for functional SDEs with multiplicative noiseand applications, Stoch. Proc. Appl. 121(2011), 2692–2710.4041] F.-Y. Wang, X. Zhang, Degenerate SDEs in Hilbert spaces with rough drifts, Infin.Dimens. Anal. Quantum Probab. Relat. Top. 18(2015), 1550026, 25 pp.[42] F.-Y. Wang, X. Zhang, Degenerate SDE with H¨older-Dini drift and non-Lipschitz noisecoefficient, to appear in SIMA J. Math. Anal.[43] L.-M. Wu, Moderate deviations of dependent random variables related to CLT, Ann.Probab. 23(1995), 420–445.[44] X. Zhang, Strong solutions of SDES with singular drift and Sobolev diffusion coeffi-cients, Stoch. Proc. Appl.