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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.

The weak convergence method of proving a large deviation principle has been developed by Dupuis and Ellis in [

However, there are still few results on the large deviation for stochastic evolution equations with jumps. In [

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 ( [

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. [

in the framework of a Gelfand’s triple:

where V, H (see Section 2) are separable Banach and separable Hilbert space respec- tively. We will establish LDP for solutions of above evolution equation on

In Section 2, we firstly give some notations and recall some results from [

We first recall some notations from [

Let

Set

Let G be a real separable Hilbert space and let Q be a positive definite and symmetric trace operator defined on G. Set

1) W is a Q-Wiener process;

2) N is a Poisson random measure with intensity measure

3)

We denote by

Denote by

where

and define a counting process

For fixed

By [

Remark 1. We note that, for

Set

We endow

Let

Let

defined on

where

Hypothesis. There exists a measurable map

1) For

where Þ denotes the weak convergence.

2) For

For

where

We have the following important result due to [

Theorem 2. Under the above Hypothesis,

Now we introduce our framework and assumptions.

Let

1)

2) V is dense in H;

3) there exists a constant c such that for all

4)

Let

where

Let

be progressively measurable. For example, for every

We assume throughout this paper that:

(H1) Hermicontinuity: For any

is continuous.

(H2) Weak monotonicity: There exist

holds on

(H3) Coercivity: For all

holds on

(H4) For all

holds on

(H5) There exists

and

(H6) There exist some compact

(H7)

Consider small noise stochastic evolution equation as following:

Under the assumptions (H1)-(H5), by [

We now fix a family of processes

By Girsanov’s theorem,

Remark 3. For

We will verify that

Lemma 1. There exists a constant

In order to characterize a compact set in

Lemma 2. For any

Proof. For fixed

Therefore

where

For

where

By (7), we have

So by (9) and dominated convergence theorem, for all

For

and

Hence, for any

By choosing

Proposition 4. For a sequence of

1) For any

2) For any

Then

Proof. It’s obvious that (2) implies the following condition (cf. [

where

For the finite family

Hence, replacing R by

Fix

Then

satisfies

By (H7), we have

It remains to prove that if a subsequence, still denoted by

According to Lemma 1 and Lemma 2, we have the following result:

Corollary 1. The sequence

Lemma 3. Assume that for almost all

Then,

Moreover, we have

and if

Proof. We divide our proof into several steps.

Step 1. By Lemma 1, we have

and

Therefore, by the strong convergence of

By (12), (16) and dominated convergence theorem, we have

Thus

Step 2. In this step, we prove

Hence, by (15) and (20), there exist subsequences of

and

Define

Note that

By taking weak limits and by (19), we can get

Indeed, for any V-valued bounded and measurable process

By (21), (23) and taking limits for

which implies

We only have to prove

Let

By (H2)

as

We now prove

Since

the last limit follows by using dominated convergence theorem. By (2), (H5), Lemma 1 and (19), we also have

and

Then limit (27) follows.

Moreover, it is easy to get that

Now we prove the following limit:

By (H5), Lemma 1 and (19), we have

where

and

For

by noting (16) and (19). For

pport

Then, we get (29).

It is obvious that

Combining (26) to (31) yields that

On the other hand, by Itô’s formula we have

So, we have

which implies (24) by (H1).

Step 3. In this step we prove (13) and (14). Notice that

By Itô’s formula, we have

where

By Lemma 1 and BDG’s inequality, we get

For

Similarly

For

Similarly

For

Assume

Set

then

So

Notice (32), we get (13) and (14) immediately.

We also have the following main lemma.

Lemma 4. There exists a probability space

nience, still denote by

1) For each

2)

3)

Moreover, we have

and if

Proof. From Corollary 1, we have

Then, the other conclusions follow from Lemma 3 and noting for

Remark 5. Assume that (H1)-(H7) and

For fixed

We point out that the difference between

Lemma 5. Assume that (H1)-(H7) and

Proof. Similar to the proofs of Lemma 1 and 2, we can get

is C-tight. As in Lemma 4, there exist a subsequence

Combining with this convergence and the method used in the proof of Lemma 3, we have

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,

where

Remark 7. If

Similar to [

The inner product in H is defined by

and the inclusions are compact.

Let

Then

Let

where

where

Consider the following stochastic porous medium equation

Let

The authors thank the Editor and the referee for their valuable comments. This work is supported in part by Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ13A010020) and the National Natural Science Foundation of China (Grant No. 11401029).

Zhao, H.Y. and Xu, S.Y. (2016) Freidlin-Wentzell’s Large Deviations for Stochastic Evolution Equations with Poisson Jumps. Advances in Pure Mathematics, 6, 676-694. http://dx.doi.org/10.4236/apm.2016.610056