Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

SummaryIn this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) ∂tvε=ℒvε+f(x,vε)+εσ(x,vε) $$\ddot W_{tx} $$ . Here ℒ is a strongly-elliptic second-order operator with constant coefficients, ℒh:=DHxx-αh, and the space variablex takes values on the unit circleS1. The functionsf and σ are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<m≦σ≦M wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv2 is aCk(S1)-valued Markov process for each 0≦κ<1/2, whereCκ(S1) is the Banach space of real-valued continuous functions onS1 which are Hölder-continuous of exponent κ. We prove, under some further natural assumptions onf and σ which imply that the zero element ofCκ(S1) is a globally exponentially stable critical point of the unperturbed equation ∂tυ0 = ℒυ0 +f(x,υ0), that υε has a unique stationary distributionvK, υ on (Cκ(S1), ℬ(CK(S1))) when the perturbation parameter ε is small enough. Some further calculations show that as ε tends to zero,vK, υ tends tovK,0, the point mass centered on the zero element ofCκ(S1). The main goal of this paper is to show that in factvK, υ is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forvK, υ is the LDP for the process υε, which has been shown in an earlier paper. Our methods of deriving the LDP forvK, υ based on the LDP for υε are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceCκ(S1) is inherently infinite-dimensional.