Large Deviations for a Reaction-Diffusion Equation with Non-Gaussian Perturbations

Abstract

In this paper we establish a large deviations principle for the non-Gaussian stochastic reaction-diffusion equation (SRDE) $\partial_t\nu^\varepsilon = \mathscr{L}\nu^\varepsilon + f(x, \nu^\varepsilon) + \varepsilon\sigma(x, \nu^\varepsilon)\ddot{W}_{tx}$ as a random perturbation of the deterministic RDE $\partial_t\nu^0 = \mathscr{L}\nu^0 + f(x, \nu^0)$. Here the space variable takes values on the unit circle $S^1$ and $\mathscr{L}$ is a strongly-elliptic second-order operator with constant coefficients. The functions $f$ and $\sigma$ are sufficiently regular so that there is a unique solution to the above SRDE for any continuous initial condition. We also assume that there are positive constants $m$ and $M$ such that $m \leq \sigma(x, y) \leq M$ for all $x$ in $S^1$ and all $y$ in $\mathbb{R}$. The perturbation $\ddot{W}_{tx}$ is the formal derivative of a Brownian sheet. It is known that if the initial condition is continuous, then the solution will also be continuous, and moreover, if the initial condition is assumed to be Holder continuous of exponent $\kappa$ for some $0 < \kappa < \frac{1}{2}$, then the solution will be Holder continuous of exponent $\kappa/2$ as a function of $(t, x).$ In this paper we establish the large deviations principle for $\nu^\varepsilon$ in the Holder norm of exponent $\kappa/2$ when the initial condition is Holder continuous of exponent $\kappa$ for any $0 < \kappa < \frac{1}{2}$, and when the initial condition is assumed only to be continuous, we establish the large deviations principle for $\nu^\varepsilon$ in the supremum norm. Moreover, we prove that these large deviations principles are uniform with respect to the initial condition.