Freidlin-Wentzell ’ s Large Deviations for Stochastic Evolution Equations with Poisson Jumps

We establish a Freidlin-Wentzell’s large deviation principle for general stochastic evolution equations with Poisson jumps and small multiplicative noises by using weak convergence method.


Introduction
The weak convergence method of proving a large deviation principle has been developed by Dupuis and Ellis in [1].The main idea is to get sevral variational representation formulas for the Laplace transform of certain functionals, and then to prove an equivalence between Laplace principle and large deviation principle (LDP).For Brownian functionals, Boué and Dupuis [2] have proved an elegant variational representation formula (also can be found in Zhang [3]).For Poisson functionals, we can see Zhang [4].Recently, a variational representation formula on Wiener-Poisson space has been established by Budhiraja, Dupuis, and Maroulas in [5].These type variational representations have been proved to be very effective for both finite-dimensional and infinite-dimensional stochastic dynamical systems (cf.[6]- [10]).The main advantages of this method are that we only have to make some necessary moment estimates.
However, there are still few results on the large deviation for stochastic evolution equations with jumps.In [11] they established the LDP by proving some exponential integrability on different spaces.Later, Budhiraja, Chen and Dupuis developed a large deviation for small Poisson perturbations of a more general class of deterministic equations in infinite dimensional ( [12]), but they did not consider the small Brownian perturbations simultaneously.Motivated by the above work, we would like to prove a Freidlin-Wentzell's large deviation for nonlinear stochastic evolution equations with Poisson jumps and Brownian motions.At the same time, nonlinear stochastic evolution equations have been studied in various literatures (cf.[13]- [17]).So we consider the following stochastic evolution equation: in the framework of a Gelfand's triple: where V, H (see Section 2) are separable Banach and separable Hilbert space respectively.We will establish LDP for solutions of above evolution equation on with the Skorokhod topology.For stochastic evolution equations without jumps, Ren and Zhang [9] and Liu [8]  applications and examples we refer the reader to [8], [9].The equations we consider here are more general than the equations considered in [11], and we use a different method.We note that, the large deviations for semilinear SPDEs in the sense of mild solutions were considered in paper [18] recently.For other recent research on this topic, see also [12], [19].
In Section 2, we firstly give some notations and recall some results from [5], which are the basis of our paper, and then introduce our framework.In Section 3, we prove the large deviation principle.In the last section, we give an application.Note that notations c, M C and below will only denote positive constants whose values may vary from line to line.

Preliminaries and Framework
We first recall some notations from [5].
Let  be a locally compact Polish space and denote by ( ) the space of all measures ν on ( ) ( ) C  be the space of continuous functions with compact support.

( )
is a Polish space endowed with the weakest topology such that for every denote by 1 P the unique probability measure on , is a Poisson random measure with intensity : Let G be a real separable Hilbert space and let Q be a positive definite and symmetric trace operator defined on G. Set  ( ) . We denote by P the unique probability measure on ( ) ( ) , Ω Ω  such that under P: We denote by t  be P-completion of the filtration t  .From now on, we will work on the probability space ( ) ( ) Denote by  the predictable σ-field on [ ] and define a counting process By [5], we can define for a function : and define ( ) We endow M S with the weak topology on the Hilbert space such that M S is a ×  with the usual product topology.Set Let  be a Polish space and let { } 0 be a set of  -valued random variables defined on ( ) ( )  is a family of measurable maps from Ω to  .Hypothesis.There exists a measurable map 0 : Ω →   such that the following hold.1) . 0 where ⇒ denotes the weak convergence.
We have the following important result due to [5].
Theorem 2. Under the above Hypothesis, { } 0 satisfies a large deviation principle with rate function I. Now we introduce our framework and assumptions.

Let ( )
, , H H ⋅ ⋅ be a real separable Hilbert space.Let V be a reflexive Banach space and * V be the dual space of V and * , V V ⋅ ⋅ denotes the corresponding dualization.
Identify H with its dual * H and the following assumptions are satisfied: 1) 2) V is dense in H; 3) there exists a constant c such that for all v V ∈ , L G H be the space of Hilbert-Schmidt linear operators from G to H, which is a real separable Hilbert space with the inner product where { } i g is an orthonormal basis of G.We denote by ( ) ; CQ L G H ∈ , and the norm : be progressively measurable.For example, for every We assume throughout this paper that: (H1) Hermicontinuity: For any , , (H6) There exist some compact Γ ⊂  , ( )

Large Deviation Principle
Consider small noise stochastic evolution equation as following: Under the assumptions (H1)-(H5), by [15], [17], there exists a unique solution in to Equation (5).By Yamada-Watanabe theorem, there exists a measurable mapping We now fix a family of processes ( )  , and put By Girsanov's theorem, Z   is the unique solution of the following controlled sto- chastic evolution equation: .
Remark 3.For ( )  , by ( 1) and ( 2), there exists a constant We will verify that   satisfies the Hypothesis with . By using the similar method as in [9], we have the following uniform estimates about Z   .Lemma 1.There exists a constant , 0 [ ] ( ) In order to characterize a compact set in Proof.For fixed 0 θ > and any t such that 0 t t T θ ≤ ≤ + ≤ , we have where :0 :0 By choosing θ and  small enough, then (10)  , there are 0 ∈  n , 2) For any Proof.It's obvious that (2) implies the following condition (cf.[20], p. 290).For any For the finite family ( ) Hence, replacing R by and 0 θ by 0 ′ ∧ θ θ in (11), we obtain that they still hold with 0 1 = n .Fix By (H7), we have It remains to prove that if a subsequence, still denoted by ( ) n X , converges in law to some X, then X is a.s.continuous.By taking the same scheme as in Proposition 3.26 (cf. [20], p. 315) and replacing d R by * V in the proof, we complete the proof.
According to Lemma 1 and Lemma 2, we have the following result: Corollary 1.The sequence { } ( ] Then,  Z solves the following equation: Moreover, we have and Proof.We divide our proof into several steps.
Step 1.By Lemma 1, we have and Therefore, by the strong convergence of ( ) Z ω as in (12).We get, for almost all ω , ( ) 0, ; L T V ; and so we have By ( 12), ( 16) and dominated convergence theorem, we have Step 2. In this step, we prove  Z solves Equation (13).By (H4) and ( 15), we have Hence, by ( 15) and ( 20), there exist subsequences of   Z , ( ) (still denoted by themselves for simplicity) and By taking weak limits and by (19), we can get Indeed, for any bounded and measurable process ξ , ( ) By ( 21), ( 23) and taking limits for 0 ↓  , then we get (see also the proof of ( 27) and (29) below) We only have to prove ) the last limit follows by using dominated convergence theorem.By ( 2), (H5), Lemma 1 and ( 19), we also have ) Moreover, it is easy to get that ( ) (28) Now we prove the following limit: By (H5), Lemma 1 and ( 19), we have ( ) , d d 0, as 0  by noting ( 16) and (19).
It is obvious that Combining (26) to (31) yields that On the other hand, by Itô's formula we have which implies (24) by (H1).
is H-valued cádlág function space respectively.In our case, there are two new difficulties.The first one is to find a sufficient condition to characterize a compact set in 4) instead of Ascoli-Arzelà's theorem for continuous case, the second one is to control the jump parts.This form of equation contains a large class of (nonliear) stochastic partial differential equation of evolutional type, for ×   through the superlinear groth of l.We can also consider the topology on M S  which makes M S  a compact space.Remark 1.We note that, for n g ,

Lemma 3 .
Assume that for almost all ω ,

1 J
, by Young inequality, we have function on [ ] 0,T ×  with the compact support [ ] 0,T × Γ , and by the weak convergence of T ϕ ν  to T ϕ ν (see Remark

T
i


not random.We have the following result.Lemma 5. Assume that (H1)-(H7) and 1 0 λ > hold.Let ( ) tight.As in Lemma 4, there exist a subsequence k m (still denoted by m) and Combining with this convergence and the method used in the proof of Lemma 3, we have result holds.Using Remark 5, Lemma 5 and Theorem 2, we obtain the following large deviation principle.Theorem 6.Under the same assumptions as in Lemma 5, { } 0 Z >   satisfies a large deviation principle with rate function I defined as in (3), i.e. for any ( ) µ  is the law of Z  in  and  is , Röckner and Zhang considered the following type semi-linear stochastic evolutions driven by Lévy processes . .
 be the law of Z  in Let ν