Limit Theorems for Large Deviations and Reaction-Diffusion Equations

Abstract

The equation $u_t = u_{xx} + u(1 - u)$ is the simplest reaction-diffusion equation. Introduction of a small parameter allows construction of geometric optics approximations for the solutions of such equations; these solutions are approximated by step-functions with the values 0 and 1. The region where the solution is close to 1 propagates according to the Huygens principle for the corresponding velocity field $v(x, e)$ which is calculated via the equation. New effects may emerge, such as stops and jumps of the wave front. The Feynman-Kac formula implies that the solutions of certain Cauchy problems obey some integral equations in the space of trajectories of the corresponding Markov processes. Examination of this equation requires the study of Laplace-type asymptotics for functional integrals. These asymptotics are defined by large deviations for the corresponding family of processes and are expressed through action functionals.