Approximation of quasi-potentials and exit problems for multidimensional RDE’s with noise

Abstract:

We deal with a class of reaction-diffusion equations, in space dimension $d>1$, perturbed by a Gaussian noise $\partial w^\delta /\partial t$ which is white in time and colored in space. We assume that the noise has a small correlation radius $\delta$, so that it converges to the white noise $\partial w/\partial t$, as $\delta \downarrow 0$. By using arguments of $\Gamma$-convergence, we prove that, under suitable assumptions, the quasi-potential $V_\delta$ converges to the quasi-potential $V$, corresponding to space-time white noise, in spite of the fact that the equation perturbed by space-time white noise has no solution.

We apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attraction of an asymptotically stable point for the unperturbed problem.