Exponential mixing for some SPDEs with Lévy noise
aa r X i v : . [ m a t h . A P ] O c t EXPONENTIAL MIXING FOR SOME SPDES WITH L ´EVYNOISE
ENRICO PRIOLA, LIHU XU, AND JERZY ZABCZYK
Abstract.
We show how gradient estimates for transition semigroups can beused to establish exponential mixing for a class of Markov processes in infinitedimensions. We concentrate on semilinear systems driven by cylindrical α -stablenoises introduced in [17], α ∈ (0 , Keywords : stochastic PDEs driven by α -stable noises, ergodicity, strong mix-ing, exponential mixing. Mathematics Subject Classification (2000) : 60H15, 47D07, 60J75, 35R60. Introduction
The paper is concerned with asymptotic properties of the stochastic evolutionequation:(1.1) dX ( t ) = [ AX ( t ) + F ( X ( t ))] dt + dZ t , X = x, driven by α -stable, α ∈ (0 , Z , introduced in [17]. The paper[17] investigated structural properties of X like strong Feller and irreducibility.Solutions turned out to be stochastically continuous but in general without c´adl´agmodifications, see [4]. The stochastic PDEs driven by L´ e vy noises have beenintensively studied since some time, see e.g. [3], [1], [16], [13], [12], [17] and [20],and the book [15] for additional references. Even for equations like (1.1), with theadditive noise, some basic questions are still open.General results on existence of invariant measures for the linear equation(1.2) dX ( t ) = AX ( t ) dt + dZ t , X = x, The first author gratefully acknowledges the support by the M.I.U.R. research project Prin2008 “Deterministic and stochastic methods in the study of evolution problems”. The secondauthor gratefully acknowledges the support from EURANDOM and Hausdorff Research Institutefor Mathematics. The third author gratefully acknowledges the support by Polish Ministry ofScience and Higher Education grant “Stochastic equations in infinite dimensional spaces” N N201419039 . with A being the infinitesimal generator of a strongly continuous semigroup, wereobtained in [6], and more recently in [18] and [10], with different assumptions on Z . The nonlinear case was investigated in [19], and by the dissipativity method in[15]; see also the recent paper [20].In the present paper we are concerned with the exponential convergence toequilibrium, called exponential mixing, see e.g. [15, Proposition 16.5] .We assumethat the nonlinearity F is Lipschitz continuous and bounded and find explicitconditions on F under which the exponential mixing takes place. The speed ofthe convergence to equilibrium can be deduced from the proofs. Our result seemsto be new even in the well-studied case when Z is a cylindrical Wiener process,corresponding to α = 2, requiring only obvious modifications of our proofs.To prove the exponential mixing we establish first gradient estimates for thetransition semigroup of the process X using the so called mild Kolmogorov equa-tion. This version of the Kolmogorov equation was introduced in [5] and elaboratedin [7] and [8]. With a similar aim gradient estimates were used in [20], in a dif-ferent framework, with the assumption α ∈ (1 ,
2) and without the exponentialbounds. The ergodicity in [20] and [21] was obtained by the technique of inter-acting particle systems called finite speed of propagation of information. For thefinite dimensional stochastic system driven by α -stable processes, [20] obtained thestrong mixing result for small F .Our main results are formulated as Theorems 2.5 and 2.6. The first one estab-lishes ergodicity without additional condition on F . Here it was enough to use themethod sketched in [9]. Theorem 2.6 is on exponential mixing with additional re-strictions on F . If the nonlinearity is only Lipschitz and bounded (and not small),it seems hard to obtain exponential mixing by the gradient estimates. We are nowworking on this general case trying a different method.The organization of the paper is as follows. Section 2 introduces the notationsand the main theorems. Section 3 proves the ergodicity and strong mixing forbounded nonlinearity. The next section is devoted to exponential gradient esti-mates. In the final section, we prove exponential mixing for small nonlinearities.2. Notations and main results
We shall study our problem in a separable Hilbert space H with an orthonormalbasis { e k } k ≥ . Denote the inner product and norm of H by h· , ·i H and | · | H re-spectively. Let C b ( H, H ) be the Banach space of all bounded continuous functions f : H → H with the supremum norm || f || := sup x ∈ H | f ( x ) | H . Similarly, C b ( H, R ) denotes the Banach space of all bounded continuous functions f : H → R with the supremum norm || f || := sup x ∈ H | f ( x ) | . XPONENTIAL MIXING FOR SOME SPDES WITH L´EVY NOISE 3
Given f ∈ C b ( H, R ) and x ∈ H , for any h ∈ H , we write(2.1) D h f ( x ) := lim ε → f ( x + εh ) − f ( x ) ε provided the the above limit exists. If h D h f ( x ) is linear and | D h f ( x ) | ≤ C | h | H , h ∈ H , then there exists a unique element in H , denoted by Df ( x ), such that D h f ( x ) = h Df ( x ) , h i H . Clearly, Df is a function from H to H . Denote by C b ( H, R ) the set of all bounded differentiable functions f : H → R with norm || f || C b := sup x ∈ H | f ( x ) | + sup x ∈ H | Df ( x ) | H . Let B b ( H, R ) be the set of all bounded (Borel) measurable functions from H into R .Further, we denote by D the set of all bounded continuous cylindrical functions f : H → R , i.e., f is some bounded continuous function depending on finitenumber of coordinates of H . For any f ∈ D , denote by Λ( f ) the localization setof f , i.e. Λ( f ) is the smallest set Λ ⊂ N such that f ∈ C b ( R Λ , R ). Finally, define D := D ∩ C b ( H, R ).Let z ( t ) be the normalized one dimensional symmetric α -stable process, 0 <α <
2, with the following characteristic function(2.2) E [ e iλz ( t ) ] = e − t | λ | α , λ ∈ R . The density of z (1) will be denoted by p α . The infinitesimal generator ∂ αx of theprocess is of the form(2.3) ∂ αx f ( x ) = 1 C α Z R f ( y + x ) − f ( x ) | y | α +1 dy with C α = − R R ( cosy − dy | y | α , see [2].We will consider equation (1.1) under the following assumptions: Assumption 2.1. • A is a dissipative operator defined by (2.4) A = X k ≥ ( − γ k ) e k ⊗ e k . < γ ≤ γ ≤ · · · ≤ γ k ≤ · · · and γ k → ∞ as k → ∞ . • F : H → H is Lipschitz and bounded . • Z = ( Z t ) is a cylindrical α -stable process with Z t = P k ≥ β k z k ( t ) e k where { z k ( t ) } k ≥ is a sequence of i.i.d. symmetric α -stable processes (defined onsome fixed stochastic basis (Ω , F , ( F t ) t ≥ , P ) with < α < and β k > , k ≥ , and (2.5) X k ≥ β αk γ k < + ∞ . E. PRIOLA, L.XU, AND J.ZABCZYK • There exists σ ∈ (0 , such that (2.6) sup k ≥ γ α − σk β k < + ∞ . Remark . Note that A generates a strongly continuous semigroup of compactcontractions ( e tA ) on H . Moreover, e tA e k = e − tγ k e k , k ≥ X =( X ( t, x )) to (1.1), see [17]. This is a predictable H -valued stochastic process,depending on x ∈ H , such that, for any t ≥ , x ∈ H , it holds ( P -a.s.): X ( t, x ) = e tA x + Z t e ( t − s ) A F ( X ( s, x )) ds + Z A ( t ) , with Z A ( t ) = Z t e ( t − s ) A dZ s . (2.7)Condition (2.6) is used in [17, Section 5] to get the strong Feller property for theMarkov semigroup P t associated with the solution X of (1.1): P t f ( x ) = E [ f ( X ( t, x ))] , f ∈ B b ( H, R ) , x ∈ H, t ≥ . For further use we need an equivalent formulation of (2.6).
Lemma 2.3.
For arbitrary σ > the following two conditions are equivalent: (2.8) B = sup k ≥ γ α − σk β k < + ∞ , (2.9) k t = sup k ≥ e − γ k t γ /αk β k ≤ ˆ c e − γ t/ t σ , t > , where ˆ c = B (2 /α ) − σ σ σ e σ . Proof.
To establish the equivalence one can argue as in the beginning of [17, Section5]. We only note the following estimate, for any n ≥ e − γ n t γ /αn β n ≤ α exp( − γ t/ e − γn t ( γ n / /α β n . (cid:3) According to [17], we have the following result for the system (1.1).
Theorem 2.4.
Under the Assumption 2.1 there exists a unique mild solution X ( t, x ) for (1.1) . Moreover its associated transition semigroup P t is strong Fellerand irreducible. This paper aims to study the long time behaviour of the system (1.1). The twomain theorems are as follows.
XPONENTIAL MIXING FOR SOME SPDES WITH L´EVY NOISE 5
Theorem 2.5.
Under Assumption 2.1 there exists a unique invariant measure µ for the system (1.1) . The measure µ is strong mixing, i.e. lim t →∞ P t f ( x ) = µ ( f ) , for all f ∈ B b ( H, R ) and x ∈ H . In the formulation of the next theorem p α stands for the density of z (1) and theconstant ˆ c was introduced in Lemma 2.3. Theorem 2.6.
Assume that Assumption 2.1 holds and that one of the followingtwo conditions holds: (i) L F < γ , where L F is the best Lipschitz constant of F ; (ii) || F || < C Γ(1 − σ ) ( γ ) − σ , where C = ˆ c R R ( p ′ α ( z )) p α ( z ) dz .Then the system (1.1) is exponentially mixing. More precisely there exist constants C = C ( | x | H , α, ( β n ) , ( γ n ) , F ) > and c = c ( α, ( β n ) , ( γ n ) , F ) > such that (2.10) | P t f ( x ) − µ ( f ) | ≤ Ce − ct || f || C b , for all f ∈ C b ( H, R ) and x ∈ H . We do not give explicit formulas for the constants C and c but they can beobtained by a careful examination of the proofs.Let us give some examples which the two main theorems can be applied to. Example 2.7.
Consider the following stochastic semilinear equation on D =[0 , π ] d with d ≥ dX ( t, ξ ) = [∆ X ( t, ξ ) + F ( X ( t, ξ ))] dt + dZ t ( ξ ) ,X (0 , ξ ) = x ( ξ ) ,X ( t, ξ ) = 0 , ξ ∈ ∂D, where Z t and F are both specified below. It is clear that ∆ with Dirichlet boundarycondition has the following eigenfunctions e k ( ξ ) = (cid:18) π (cid:19) d sin ( k ξ ) · · · sin ( k d ξ d ) , k ∈ N d , ξ ∈ D. It is easy to see that ∆ e k = −| k | e k , i.e. γ k = | k | = k + . . . + k d , for all k ∈ N d .We study the dynamics (2.11) in the Hilbert space H = L ( D ) with orthonormalbasis { e k } k ∈ N d . Z = ( Z t ) is some cylindrical α -stable noises which, under the basis { e k } k , is defined by Z t = X k ∈ N d | k | β z k ( t ) e k where { z k ( t ) } k are i.i.d. symmetric α -stable processes with α ∈ (0 ,
2) and β a realnumber. Note that P k ∈ N d | k | βα | k | < ∞ if and only if 2 > d + αβ . E. PRIOLA, L.XU, AND J.ZABCZYK
From Theorems 2.5 and 2.6, we have • If F is a bounded Lipschitz function and2 > d + αβ, α − β < , or equivalently, dα < α − β < , then the system (2.11) is strongly mixing. • If in addition || F || is sufficiently small then the system (2.11) is exponen-tially mixing. 3. Proof of Theorem 2.5
According to Sections 5.2 and 5.3 of [17], the system (1.1) is irreducible andstrong Feller. By Doob’s theorem (see [8, Theorem 4.2.1]), to prove Theorem2.5, one only needs to show the existence of invariant measures. To this purposeit is enough to establish the tightness of {L ( X ( t, x )) } t ≥ for some x ∈ H (here L ( X ( t, x )) denotes the law of the random variable X ( t, x ), see (2.7)). For this webasically follow [9].Fix any x ∈ H and set X ( t, x ) = X ( t ). First note that for any fixed t > e At is a contraction and a compact operator. Contraction property is clear bythe assumption of A . The compactness is an easy corollary of the fact that theeigenvalues − γ k tend to −∞ .According to [17, Proposition 4.2], for any (small) ε >
0, there exists some M = M ǫ > P {| Z A ( t ) | H ≤ M } ≥ − ε uniformly for all t ≥
0. By the assumptions on F and A , one clearly has | R t e A ( t − s ) F ( X ( s )) ds | H ≤ C uniformly for t ≥ ω ∈ Ω. Hence,(3.1) P {| X ( t ) | H ≤ | x | + C + M } ≥ P {| Z A ( t ) | H ≤ M } ≥ − ε. Let us rewrite (2.7) as X ( t ) = e A X ( t −
1) + Z tt − e A ( t − s ) F ( X ( s )) ds + Z tt − e A ( t − s ) dZ s , thanks to the compactness of e A and (3.1), the family L ( { e A X ( t − } ) t ≥ is tight.The integrals R tt − e A ( t − s ) dZ s have the same law as Z A (1) for all t ≥
1, and thustheir laws are of course tight.To complete the proof of tightness of {L ( X ( t )) } t ≥ , it is enough to show thatthe values of the integrals R tt − e A ( t − s ) F ( X ( s )) ds are contained in a compact set,for any t ≥
1. To this purpose note that the operator R from L (0 , H ) into H R φ = Z e As φ ( s ) ds, φ ∈ L (0 , H ) , XPONENTIAL MIXING FOR SOME SPDES WITH L´EVY NOISE 7 is compact (this follows from the compacteness of the operators e At , t > R and the fact that the transformation F is bounded, we getthat the integrals are contained in a fixed compact set. The proof is complete.4. Exponential gradient estimates
To establish Theorem 2.6 we first derive exponential gradient estimates for theGalerkin approximation of the equation (1.1).4.1.
Galerkin approximation.
Let { e k } k ≥ be the orthonormal basis associatedwith the operator A and write Γ N := { , . . . , N } For any x ∈ H and any integer N > ( dX Nk ( t ) = [ − γ k X Nk ( t ) + F Nk ( X N ( t ))] dt + β k dz k ( t ) ,X Nk (0) = x k , for all k ∈ Γ N , where x N = ( x k ) k ∈ Γ N , x k = h x, e k i H , and F Nk ( x N ) = h F ( x N , , e k i H .Eq. (4.1) can be written in the following vector form(4.2) ( dX N ( t ) = [ AX N ( t ) + F N ( X N ( t ))] dt + dZ Nt ,X N (0) = x N , where X N ( t ) = ( X Nk ( t )) k ∈ Γ N and Z Nt = ( β k z k ( t )) k ∈ Γ N . The infinitesimal generatorof (4.2) is L N = X k ∈ Γ N β αk ∂ αk + X k ∈ Γ N [ − γ k x k + F Nk ( x N )] ∂ k = X k ∈ Γ N [ β αk ∂ αk − γ k x k ∂ k ] + X k ∈ Γ N F Nk ( x N ) ∂ k , (4.3)where ∂ k = ∂ x k and ∂ αk = ∂ αx k . In the sequel we will identify x N ∈ R N with x N = N X k =1 x k e k ∈ H. Consider the Kolmogorov equation of the Galerkin approximation(4.4) ( ∂ t u N ( t ) = L N u N ( t ) ,u N (0) = f, where f ∈ D with Λ( f ) ⊂ Γ N . According to Section 5.3 of [17], Eq. (4.4) has amild solution P Nt f which satisfies(4.5) P Nt f ( x N ) = S Nt f ( x N ) + Z t S Nt − s [ h F N , DP Ns f i H ]( x N ) ds, E. PRIOLA, L.XU, AND J.ZABCZYK where S Nt is the Ornstein Uhlenbeck transition semigroup generated by the oper-ator P k ∈ Γ N [ β αk ∂ αk − γ k x k ∂ k ]. Moreover, we also have P Nt f ( x N ) = E [ f ( X N ( t, x N ))] . ¿From the second step of the proof of [17, Theorem 5.7], we have(4.6) lim N →∞ P Nt f ( x N ) = P t f ( x ) , in particular, for any f ∈ D with Λ( f ) ⊂ Γ N , t ≥ , x ∈ H , where P t is thetransition semigroup defined in Theorem 2.4.4.2. Estimates and their proofs.
The gradient estimates which are establishedhere are of two different types. The first one is straightforward, and althoughformulated for (4.1), is true in a much more general situation. Moreover, it isstated in terms of the best Lipschitz constant of the nonlinearity F . The secondone is true for the specific systems considered in the paper, requires more subtleconsiderations and is stated in terms of the supremum of {| F ( x ) | ; x ∈ H } . Proposition 4.1.
Let P Nt be the transition semigroup corresponding to the solu-tion of the equation (4.1). Then (4.7) || DP Nt f || ≤ e − ( γ − L F ) t || Df || , f ∈ D with Λ( f ) ⊂ Γ N , t ≥ . Proof.
Denoting by X N ( t, x N ) the solution of Eq. (4.1) with initial data x N , onehas | X N ( t, x N ) − X N ( t, y N ) | H ≤ e − ( γ − L F ) t | x N − y N | H , t ≥ , x, y ∈ H, which implies lim ε → sup | X N ( t,x N + εh N ) − X N ( t,x N ) ε | ≤ e − ( γ − L F ) t | h N | , h ∈ H . Hence,by the dominated convergence theorem one has | D h P Nt f ( x N ) | ≤ E (cid:20) | Df ( X N ( t )) | H lim ε → sup (cid:12)(cid:12)(cid:12)(cid:12) X N ( t, x N + εh N ) − X N ( t, x N ) ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ≤ e − ( γ − L F ) t || Df || | h | H . (cid:3) Proposition 4.2.
Let P Nt be the transition semigroup corresponding to the so-lution of the equation (4.1). There exists a positive constant C , depending on α, ( β n ) , ( γ n ) , σ , such that || DP Nt f || ≤ Ce − ωt || Df || , f ∈ D , with Λ( f ) ⊂ Γ N , t ≥ , where ω = γ − ( C k F k Γ(1 − σ )) − σ , and the constant C was introduced in Theorem 2.6. To prove Proposition 4.2, we need the following lemma.
XPONENTIAL MIXING FOR SOME SPDES WITH L´EVY NOISE 9
Lemma 4.3.
For any f ∈ D , with Λ( f ) ⊂ Γ N , one has: ( i ) | DS Nt f ( x N ) | H ≤ e − γ t || Df || , ( ii ) | DS Nt f ( x N ) | H ≤ C e − γ t/ t σ || f || , for all x ∈ H , N ≥ , t > (where C = ˆ c R R ( p ′ α ( z )) p α ( z ) dz , see Theorem 2.6).Proof. Note that, for any vector h ∈ P Nk =1 h k e k , x ∈ H, we have h DS Nt f ( x N ) , h i = h DS t f ( x N ) , h i , where S t is the Ornstein-Uhlenbeck semigroup acting on C b ( H, R ) associated to X ( t, x ) when F = 0. We know that(4.8) S t f ( x ) = Z H f ( e tA x + y ) µ t ( dy ) , where µ t is the law of the random variable Z A ( t ) = R t e ( t − s ) A dZ s . By differentiatingunder the integral sign in (4.8) we immediately get the first assertion (remark that k e tA k ≤ e − γ t , t ≥
0, where k e tA k denotes the operator norm of e tA ).As for the second assertion, recall the gradient estimate from [17, Theorem 4.14] | DS t f ( x ) | H ≤ c α (cid:16) sup k ≥ e − γ k t γ /αk β k (cid:17) || f || , f ∈ C b ( H, R ) , t > , where c α = 18 Z R ( p ′ α ( z )) p α ( z ) dz. According to (2.9) we have | DS t f ( x ) | H ≤ c α ˆ c e − γ t/ t σ || f || , t > , (4.9)and the assertion follows. (cid:3) Proof of Proposition 4.2.
By (4.5) and Lemma 4.3, we have, for any N ∈ N , | DP Nt f ( x N ) | ≤ e − γ t || Df || + Z t C e − γ ( t − s ) ( t − s ) σ ||h F N , DP Ns f i H || ds, where C is defined in Lemma 4.3; therefore, || DP Nt f || ≤ e − γ t || Df || + Z t C e − γ ( t − s ) ( t − s ) σ || F || || DP Ns f || ds, Writing v N ( t ) = e γ t || DP Nt f || , we have from the above inequality v N ( t ) ≤ || Df || + Z t C || F || ( t − s ) σ v N ( s ) ds. Now we use the following Henry’s estimate (see [11]): let a ≥ b ≥ β > u on [0 , T ) such that u ( t ) ≤ a + b Z t ( t − s ) β − u ( s ) ds, t ∈ [0 , T ) , then we have u ( t ) ≤ aG β ( θt ), t ∈ [0 , T ), where θ = ( b Γ( β )) /β , G β ( z ) = X n ≥ z nβ Γ( nβ + 1) , z ≥ G β ( z ) ∼ β e z as z → + ∞ ).In our case u ( t ) = v N ( t ), a = k Df k , b = C k F k and β = 1 − σ . Thus for aconstant C , depending on σ , and all positive t we get v N ( t ) ≤ C k Df k − σ exp n ( C k F k Γ(1 − σ )) − σ t o Therefore || DP Nt f || ≤ C k Df k − σ exp n ( C k F k Γ(1 − σ )) − σ t − γ t o and we get the assertion. (cid:3) Proof of Theorem 2.6
From Theorem 2.5, the system (1.1) is ergodic and has a unique invariant mea-sure µ .Note that it is enough to prove (2.10) for any f ∈ D . Indeed then, approx-imating any function f ∈ C b ( H ) by a sequence ( f n ) ⊂ D such that f n → f and Df n → Df pointwise with sup n ≥ k f n k C b < ∞ , we get easily the completeassertion. Let us fix f ∈ D and suppose that Λ( f ) ⊂ Γ N .The crucial point of the proof is to apply the gradient estimate from Proposition4.2, in the spirit of [15, Proposition 16.4].Concerning the case (i), by using (4.7), we can show the exponential mixing bya similar argument as for the case (ii). So from now on we shall concentrate onthe proof of the case (ii).First let us suppose that α ∈ (1 , Z NA ( t ) = Z t e A ( t − s ) dZ Ns and write Y N ( t ) = X N ( t, x N ) − Z NA ( t ) so that Y N ( t ) = e At x N + Z t e A ( t − s ) F N ( X N ( s, x N )) ds. For any t > t >
0, writing s = t − t , by the gradient estimates of Proposition4.2, we have, for any x ∈ H , XPONENTIAL MIXING FOR SOME SPDES WITH L´EVY NOISE 11 | P Nt f ( x N ) − P Nt f ( x N ) | = | E [ P Nt f ( X N ( s, x N )) − P Nt f ( x N )] |≤ || DP Nt f || E | X N ( s, x N ) − x N | H ≤ Ce − ωt || Df || E | X N ( s, x N ) − x N | H , (5.1)where C, ω > | X N ( t, x N ) − x N | H ≤ | e At x N − x N | H + (cid:12)(cid:12) Z t e A ( t − s ) F N ( X N ( s, x N ) ds (cid:12)(cid:12) H + | Z A ( t ) | H . We have (cid:12)(cid:12) Z t e A ( t − s ) F N ( X N ( s, x N )) ds (cid:12)(cid:12) H ≤ k F k Z ∞ e − γ s ds ≤ k F k γ , for any ω ∈ Ω, x ∈ H and t ≥
0. Concerning | Z A ( t ) | H remark that since α > E | Z A ( t ) | H ≤ ˜ c (cid:16) X n ≥ | β n | α (1 − e − αγ n t ) αγ n (cid:17) /α ≤ ˜ c (cid:16) X n ≥ | β n | α αγ n (cid:17) /α < + ∞ , t ≥ . It follows that E | X N ( s, x N ) − x N | H ≤ C (1 + | x N | H ) , where C does not depend on N , s ≥ x ∈ H .By (5.1) we get | P Nt f ( x N ) − P Nt f ( x N ) | ≤ C e − ωt || Df || (1 + | x N | H ) . Passing to the limit as N → ∞ , we get (see the proof of Theorem 5.7 in [17]) | P t f ( x ) − P t f ( x ) | ≤ C e − ωt || Df || (1 + | x | H ) . where C does not depend on t . This estimate shows that P t f ( x ) converges tosome constant exponentially fast as t → + ∞ . By ergodicity of the system, thisconstant must be µ ( f ).Let us consider now α ∈ (0 , f ∈ C b ( H, R ), s ∈ (0 , f ] s by[ f ] s := sup x = y | f ( x ) − f ( y ) || x − y | s . We have(5.2) [ f ] s ≤ − s sup x = y | f ( x ) − f ( y ) | s | x − y | s || f || − s ≤ − s || f || − s || Df || s . Now we choose p ∈ (0 , α ). Using that E | Z A ( t ) | pH < ∞ (see [17, Theorem 4.4]) and(5.2) with s = p we get, arguing as before, | P Nt f ( x N ) − P Nt f ( x N ) | = | E [ P Nt f ( X N ( s, x N )) − P Nt f ( x N )] |≤ − p || DP Nt f || p k f k − p E | X N ( s, x N ) − x | pH ≤ − p C p e − ωp t || f || C b E | X N ( s, x N ) − x N | pH . Since E | Z A ( t ) | pH ≤ ˜ c p (cid:16) X n ≥ | β n | α (1 − e − αγ n t ) αγ n (cid:17) p/α ≤ ˜ c p (cid:16) X n ≥ | β n | α αγ n (cid:17) p/α < + ∞ , t ≥ , it follows that E | X N ( s, x ) − x N | pH ≤ C (1 + | x N | H ) p , where C does not depend on N , s ≥ x ∈ H . Finally we have | P t f ( x ) − P t f ( x ) | ≤ C e − ωp t || f || C b (1 + | x | H ) p , where C does not depend on t . Arguing as before we complete the proof. Acknowledgements.
The authors thank the Newton Institute (Cambridge), wherethis paper was initiated, for its hospitality.
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