The generalized Ginzburg-Landau equation, as a basic model of nonlinear phenomena, is used in various areas of the physics, including Rayleigh-Benard convection,the Taylor-Couette flow in fluid mechanics, the drift dissipative wave in plasma physics, and turbulent flows in chemical reactions (see e.g., [3, 4, 9, 12, 14, 16, 19, 20]).For the stochastic Ginzburg-Landau equation (SGLE for short), a long list of studies on existence and uniqueness, as well as the asymptotic behavior of solutions, have appeared; see [2, 21, 30, 35, 36] and the references therein.In particular, Yang [36] proved the existence and uniqueness of the solution for the two-dimensional SGGLE with a multiplicative noise,and Lin and Gao[21]studied the SGGLE driven by jumps.

The objective of this paper is to study the large deviation principle (LDP for short) for uεgiven in eq.(1.1).First, let us rewrite (1.1) in an abstract form.

By using the argument in the proof of Theorem 2.5 in Lin and Gao [21], one can obtain the following well-posedness result for eq.(1.4):

There are some references about LDPs for the stochastic partial differential equations(SPDEs for short) of Ginzburg-Landau’s type; see e.g., Yang and Hou [37] for the stochastic Ginzburg-Landau equation with multiplicative Gaussian noise, Yang and Pu [38] for the stochastic cubic Ginzburg-Landau equation, and Pu and Huang [26] for the two-dimensional derivative Ginzburg-Landau equation.In their proofs, the weak convergence method based on a variational representation for positive measurable functionals of the Brownian motion played an important role (see e.g., [6, 7]).

The weak convergence criterion of the LDP in the case of Poisson random measures was introduced by [5, 8].Recently, a new sufficient condition for verifying the large deviation criterion of Budhiraja,Dupuis and Maroulas[7]was given by Matoussi,Sabbagh and Zhang[25].This was used by Liu, Song, Zhai and Zhang in [23] to study the LDP for the Mckean-Vlasov equation with jumps.The advantage of this new sufficient condition is to avoid proving the tightness of the controlled stochastic partial differential equation.This new sufficient condition was recently applied to the study of LDPs in [10, 31, 32].

There are many results about LDPs related to SPDEs driven by jump noise (see e.g.,[5, 11, 27, 28, 33, 34, 39–41]).Among these is the work of Xiong and Zhai [33], who provided a unified proof of LDPs for a large class of SPDEs with locally monotone coefficients driven by L′evy noise using the weak convergence approach.However, the method in [33] does not work for the SGGLE (1.4), since the local monotonicity conditions(i)–(iv)in[33]cannot be satisfied in a straight way.Although we can use the nonlinear structure and use the argument in [21]to obtain the local monotonicity condition (ii), the coercivity (iii) and the growth condition (iv)are very hard to verify for the SGGLE (1.4).To overcome this diffculity, we use the argument of [21] to get some more precise estimates in the study of the LDP for (1.4).

The rest of this paper is organized as follows: in Section 2, we first recall the Poisson random measure and the weak convergence criterion for the LDP obtained in [7] and [25], and then we present the main result of this paper.Section 3 is devoted to studying the skeleton equation.In Sections 4 and 5, we verify the two conditions for the weak convergence criterion.

2 Preliminaries and Main Results

2.1 Poisson Random Measure

Recall that Z is a locally compact Polish space.Denote by MFC(Z) the collection of all measures on (Z,B(Z)) such that ν(K)<∞for any compact K ∈B(Z).Denote by Cc(Z) the space of the continuous functions with compact supports,endowing MFC(Z)with the weakest topology such that, for every f ∈Cc(Z), the function

is continuous for every ν ∈MFC(Z).This topology can be metrized such that MFC(Z) is a Polish space (see e.g., [8]).

For any T ∈(0,∞), we denote that ZT= [0,T]×Z and νT= λT?ν, with λTbeing the Lebesgue measure on [0,T] and ν ∈MFC(Z).Lettingnbe a Poisson random measure on ZTwith intensity measure νT, it is well-known ([17]) thatnis an MFC(ZT) valued random variable such that

(i) for any B ∈B(ZT) with νT(B)<∞,n(B) is Poisson distributed with mean νT(B);

(ii) for any disjoint sets B1,···,Bk∈B(ZT),n(B1),···,n(Bk) are mutually independent random variables.

For notational simplicity, we write, from now on, that

and denote by P the probability measure induced bynon (M,B(M)).Under P, the canonical map, η : M →M, η(m).= m is a Poisson random measure with intensity measure νT.With applications to large deviations in mind,we also consider,for θ >0,probability measures Pθon(M,B(M)) under which η is a Poisson random measure with intensity θνT.The corresponding expectation operators will be denoted by E and Eθ, respectively.

Denote that

Here η?is called a controlled random measure, with ? selecting the intensity for the points at location x and time s, in a possibly random but non-anticipating way.When ?(s,x, ˉm)≡θ ∈(0,∞), we write η?= ηθ.Note that ηθhas the same distribution with respect to ˉP as η with respect to Pθ.

2.2 A General Criterion for Large Deviation Principle

Let {uε}ε>0be a family of random variables defined on a probability space (?,F,P) and taking values in a Polish space E.

Definition 2.1(Rate function) A function I : E →[0,∞] is called a rate function on E if, for each M <∞, the level set {y ∈E :I(y)≤M} is a compact subset of E.

Definition 2.2(Large deviation principle) Let I be a rate function on E.The sequence{uε}ε>0is said to satisfy the LDP on E with the rate function I if the following two conditions hold:

(a) for each closed subset F of E,

Let {Gε}ε>0be a family of measurable maps from M to U, where M is given by (2.1) and U is a Polish space.We present a sufficient condition established in [23, 25] to obtain an LDP of the family Gε(εηε?1) as ε →0.

Condition 2.3Suppose that there exists a measurable map G0: M →U such that the following two items hold:

(A) for any N ∈N, let ?n, ? ∈SNbe such that ?n→? as n →∞.Then

By convention, I(φ) =∞if Sφ=?.The following theorem was proved in Theorem 3.2 of [25]and Theorem 4.4 of [23]:

2.3 Main Result

Let uεbe the solution to eq.(1.4).It follows from Theorem 1.3 that, for every ε>0, there exists a measurable map Gε: ˉM →D([0,T];H) such that, for any Poisson random measurenε?1on[0,T]×Z with intensity measure ε?1λT?ν given on some probability space,Gε（εnε?1）is the unique solution of(1.4)with ?ηε?1replaced by ?nε?1;here ?nε?1is the compensated Poisson random measure ofnε?1.

ProofAccording to Theorem 2.4, it is sufficient to prove that Condition 2.3 is fulfilled.The verification of Condition 2.3 (A) will be given by Proposition 4.1.Condition 2.3 (B) will be verified in Proposition 5.3.The proof of Theorem 2.6 is complete.?

3 The Skeleton Equation

In this section,we prove the following results of the solution to the skeleton equation(2.8):

3.1 Some Lemmas

Lemma 3.2Suppose that Conditions 1.1 and 2.5 hold.Then we have the following results:

(1) ([5, Lemma 3.4]) for every N ∈N,

endowed with the natural norm.Then the embedding of Λ in Lp([0,T];H) is compact.

3.2 Finite-Dimensional Approximations

We use the Galerkin method to prove the existence of the solution to (2.8).

Suppose that {ei:i ∈N}?D(A) is an orthonormal basis of H such that the span{ei:i ∈N} is dense in V.Denote that Hn:= span{e1,···,en}.Let Pnbe the orthogonal projection onto Hnin H, i.e.,

For simplicity, we denote that

where C3,8= C3,8(T,λ1,λ2) ∈(0,∞).By using the same argument as to that in the proof of(3.17), we have that, for any δ ∈(0,), there exists a constant L4>0 such that

The proof is complete.?

3.3 Proof of Proposition 3.1

In this part, we prove Proposition 3.1, which is inspired by Section 3 of [21].

By (3.3) and (3.4) in Lemma 3.2, we know that (3.26) holds.

Next we prove (3.27).Take ξ ∈L2((0,T];H2)∩L∞([0,T];V)∩L2σ+2([0,T];L2σ+2(D)).By Lemmas 3.5 and 3.6 in [21], we have that, for any ε1,ε2>0,

4 Verification of Condition 2.3 (A)

Combining (4.7) and (4.8), we obtain (4.5).

Recall r defined by (3.31).By using the same argument as in the proof of (3.27), to prove(4.6), it suffices to prove that

A straightforward calculation gives that

where C4,4∈(0,∞).By using the same method as to that in the proof of (3.26), we obtain that, for any ε>0,

where M is defined by (3.28).Since ε is arbitrary, by Lemma 3.2 and (4.1), we have that

which converges to 0, as n →∞, by (4.11).

The proof is complete.?

5 Verification of Condition 2.3 (B)

Recall ?ANfor any N ≥1 in Subsection 2.2.For any ?ε∈?AN, consider the controlled SPDE

which completes the proof of this proposition.?