Moderate deviation principle for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction

ABSTRACT In this article, we obtain a central limit theorem and prove a moderate deviation principle for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term.

For each i = 1, . . . , r, where the coefficients a i hk are taken in C ∞ (Ō), the matrices a i (ξ ) := [a i hk (ξ )] hk are non-negative and symmetric for each ξ ∈Ō and fulfil a uniform ellipticity condition, i.e. inf ξ ∈Ō d h,k=1 a i (ξ )v h v k ≥ λ i |v| 2 , v ∈ R d , for some positive constants λ i , and coefficients b i h are continuous. The operators B i act on the boundary of O and are assumed to be either of Dirichlet or of co-normal type. The linear operators Q j are bounded on L 2 (O) and may be equal to the identity operator if d = 1. Let ( , F, P) be a probability space with an increasing family {F t } 0≤t≤T of the sub-σ -fields of F satisfying the usual conditions. The noises ∂ω j ∂t are independent cylindrical Wiener processes on ( , F, F t , P). Cramér (1938) established expansions of tail probabilities about sums of independent random variables defying the normal distribution. Comparing to large deviation, which offers a accurate estimate associated with the law of large number, moderate deviation is commonly applied to provide further estimates concerning the central limit theorem and the law of iterated logarithm.
Moderate deviation is an intermediate estimation between the large deviation with deviation scale λ(ε) = 1/ √ ε and the CLT with λ(ε) = 1. The MDP provides rates of convergence and a way to construct asymptotic confidence interval, see Gao and Zhao (2011) and the references therein. In recent years, there is an increasing interest on the study of MDP. For independent and identically distributed random sequences, Chen (1990Chen ( , 1991Chen ( , 1993 and Ledoux (1992) found the necessary and sufficient conditions for MDP. Moderate deviation was discussed by Djellout and Guillin (2001), Gao (1996Gao ( , 2003, and Wu (1995Wu ( , 1999 for Markov processes as well as Guillin and Liptser (2006) for diffusion processes. For mean field interacting particle models, MDP was worked by Douc et al. (2001).  and  discussed moderate deviations for stochastic reaction-diffusion equation and 2D stochastic Navier-Stokes equations, respectively. Recently, there are several works on moderate deviations for SPDEs with jump, see Budhiraja et al. (2016) and Dong et al. (2017).
In this paper, we prove a CLT and establish a MDP for the class of reaction-diffusion systems of the form (1). Since we consider the solutions in spaces of continuous functions and there is no Itô formula, we will work directly on heat kernels. Also, because of the weak assumptions on drift and diffusion coefficients, we have to deal with the estimates on the solutions delicately.
The organization of this paper is as follows. The assumptions will be given in Section 2. Section 3 is devoted to prove the CLT. In Section 4, we establish the MDP.
Throughout this paper, c and c n are positive constants independent of ε and those values may be different from line to line.

Preliminaries
In this section, we formulate the equation and state the precise conditions on the coefficients.
Let H be the separable Hilbert space L 2 (O; R r ) endowed with the scalar product |x| H := r i=1 |x i | L 2 (O) . We shall denote by A the realization in H of the differential operator A := (A 1 , . . . , A r ), endowed with the boundary conditions B := Davies (1989), it is proved that C is a non-positive and self-adjoint operator which generates an analytic semigroup e tC with dense domain given by e tC = (e tC 1 , . . . , e tC r ), where C i is the realization in L 2 (O) of C i with the boundary condition B i . Moreover, denoted by L p the realization of the operator L in L p (O; R r ) for any p ≥ 1. We see that L p x = L q x, x ∈ D(L p ) ∩ D(L q ) and denote all of them by L if there is no confusion.
The covariance operator Q of the Wiener process W(·) is a positive symmetric, trace class operator on H.
The hypothesis concerns the eigenvalues of A: Hypothesis 2.1: The complete orthonormal system of H which diagonalizes A is equi-bounded in the sup-norm.
Assume that Q := (Q 1 , . . . , Q r ) : H → H is a bounded linear operator which satisfies: Hypothesis 2.2: Q is non-negative and diagonal with respect to the complete orthonormal basis which diag- The assumptions on the coefficients f and g are as follows.

Hypothesis 2.3: The mapping
We set for any x, y ∈ E and t ≥ 0, , and for f = (f 1 , . . . , f r ), define for any t ≥ 0 the composition operator F(t, ·) by setting, for any (3) One of the following two conditions holds: Then the system (1) can be rewritten in the following type: where the operator A can be written as A = C + L.

Central limit theorem
In this section, our task is to establish the CLT. We will need some results in Cerrai (2003), and we list them here for the convenience of readers.
Proof: Define It is immediate to check that F n is Lipschitz-continuous in E. Then if we consider and du 0 there exist unique mild solutions u ε n and u 0 n in L p ( ; C((s, T]; E) ∩ L ∞ (s, T; E)) for Equations (14) and (15), respectively. Define Therefore, due to (10), Next we can repeat the same arguments in the intervals [s + t 0 , s + 2t 0 ], [s + 2t 0 , s + 3t 0 ] and so on and for any T > s, with the help of (11) for u ε n and u 0 n .
Let V ε (t) := (u ε − u 0 )/ √ ε be the solution of the following SPDE: with initial value V ε (0) = 0. Let V 0 be the solution of the following SPDE: where we need an additional assumption about F .

Hypothesis 3.1:
We now come to the main result of this section.

Theorem 3.4 (Central Limit Theorem):
Under Hypotheses 2.1-3.1, V ε (t) converges to V 0 in the space L p ( ; C([0, T]; E)) in probability, that is, there exists a constant ε 0 > 0 such that, for all 0 < ε ≤ ε 0 , and we see that F n is Lipschitz-continuous in E. Then if we consider There exists unique mild solutions V ε n and V 0 n in L p ( ; C([0, T]; E)), respectively. Define the following mappings Due to inequality (16) If we take t 1 > 0 such that c ψ s (s + t 1 ) < Consider There exists a random field η ε (r, x) taking values in (0, 1) such that Then, In view of last inequality, the Lipschitz continuity of F n and Hypothesis 3.1, it yields Inequalities (20) according to inequality (17). Also, exp{ s+t 1 s 2c n (s + t 1 ) sup r ∈[s,r] (F 0 |u 0 (r )| E + F 1 ) dr} is finitely controlled by some positive constant M from |u 0 n | E ≤ c s,2p (s + t 1 )(1 + |x| E ).
Firstly we recall the general criteria for an LDP given in Budhiraja and Dupuis (2000). Let E be a Polish space with the Borel σ -field B(E). For any h ∈ H 0 , consider the deterministic equation with initial value X h (0) = 0 and for any φ ε ∈ A, consider Now we are ready to state the second main result.
We will adopt the following weak convergence method to prove the MDP.
Then the family {Y ε } ε>0 satisfies an LDP in E with the rate function I given by with the convention inf{∅} = ∞.
To prove Theorem 4.3, we only need to verify the following two propositions according to Theorem 4.4.
Proposition 4.5: Under the same conditions as Theorem 4.3, for every fixed N ∈ N, let φ ε , φ ∈ A N be such that φ ε converges in distribution to φ as ε → 0. Then ε (W(·) + λ(ε)  We start to prove Proposition 4.5 and we need the following lemma and inequalities.
In view of the Lipschitz continuity of F n and inequality (16)