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

Abstract

Summary. In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) ~?tv~ = 5or ~ + f (x, v ~) + ca(x, v~) Wtx. Here 5O is a strongly-elliptic second-order operator with constant coefficients, 5~DHxx-~xh, and the space variable x takes values on the unit circle S 1. The functionsfand a are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0 < m _< o-< M where m and M are some finite positive constants. The perturbation W is a Brownian sheet. It is well-known that under some simple assumptions, the solution v ~ is a Ck(S1)-valued Markov process for each 0 __< tc < 1/2, where CK(S 1) is the Banach space of real-valued continuous functions on S ~ which are H61der-continuous of exponent ~c. We prove, under some further natural assumptions onfand a which imply that the zero element of C ~(S 1) is a globally exponentially stable critical point of the unperturbed equation c~tv ~ = 5or ~ +f(x, v~ that v ~ has a unique stationary distribution v ~'~ on (C"(S1), N(C"(S~))) when the perturbation parameter e is small enough. Some further calculations show that as e tends to zero, v K'~ tends to v ~' o, the point mass centered on the zero element of C"(S1). The main goal of this paper is to show that in fact v ~'* is governed by a large deviations principle (LDP). Our starting point in establishing the LDP for v ~' ~ is the LDP for the process v ~, which has been shown in an earlier paper. Our methods of deriving the LDP for v"' ~ based on the LDP for v ~ are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state space C"(S ~) is inherently infinite-dimensional. Mathematics Subject Classifications (1985)" 60F10, 60H15* This work was performed while the author was with the

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