LARGE DEVIATIONS FOR THE INVARIANT MEASURE OF A REACTION-DIFFUSION EQUATION WITH NON-GAUSSIAN PERTURBATIONS

In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) partial derivative t-upsilon(epsilon) = L-upsilon(epsilon) + f(x, upsilon(epsilon)) + epsilon-sigma(x, upsilon(epsilon)) W(tx). Here L is a strongly-elliptic second-order operator with constant coefficients, Lh:= DH(xx) - alpha-h, and the space variable x takes values on the unit circle S1. The functions f and sigma are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0 < m less-than-or-equal-to sigma less-than-or-equal-to 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 upsilon(epsilon) is a C(k)(S1)-valued Markov process for each 0 less-than-or-equal-to kappa < 1/2, where C(kappa)(S1) is the Banach space of real-valued continuous functions on S1 which are Holder-continuous of exponent kappa. We prove, under some further natural assumptions on f and sigma which imply that the zero element of CK(S1) is a globally exponentially stable critical point of the unperturbed equation partial derivative t-upsilon-0 = L-upsilon-0 + f(x, upsilon-0), that upsilon(epsilon) has a unique stationary distribution nu(kappa,epsilon) epsilon on (C(kappa)(S1), B(C(kappa)(S1))) when the perturbation parameter epsilon is small enough. Some further calculations show that as epsilon tends to zero, nu(kappa,epsilon) tends to nu(kappa,0), the point mass centered on the zero element of C(kappa)(S1). The main goal of this paper is to show that in fact nu(kappa,epsilon) is governed by a large deviations principle (LDP). Our starting point in establishing the LDP for nu(kappa,epsilon) is the LDP for the process upsilon(epsilon), which has been shown in an earlier paper. Our methods of deriving the LDP for nu(kappa,epsilon) based on the LDP for upsilon(epsilon) are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state space C(kappa)(S1) is inherently infinite-dimensional.