Stochastic Hamilton-Jacobi equations

Summary

Abstract

In this paper we describe the stochastic Hamilton Jacobi theory and its applications to stochastic heat equations, Schrodinger equations and stochastic Burgers’ equations.

Keywords. Stochastic Hamilton Jacobi equation, Stochastic heat equation, Stochastic Burgers’ equation, Stochastic harmonic oscillator.

Introduction

It is well-known that the leading term in Varadhan's and Wentzell-Freidlin's large deviation theories (Varadhan (1967), Wentzell and Freidlin (1970)) and Maslov's quasi-classical asymptotics of quantum mechanics (Maslov (1972)) involves the solution of a variational problem which gives a Lipschitz continuous solution of the Hamilton Jacobi equation if it exists (Fleming (1969, 1986)). It was proved by Truman (1977) and Elworthy and Truman (1981, 1982) that before the caustic time the Hamilton Jacobi function which is C^1,2 gives the exact solutions of the diffusion equations geared to small time asymptotics. The main tools in the theory are classical mechanics and the Maruyama-Girsanov-Cameron-Martin formula. The philosophy of the theory is to choose a suitable drift for a Brownian motion on the configuration space manifold and to employ the MGCM theorem to simplify the Feynman-Kac representation of the solutions for the heat equations. An extended version of this theory to degenerate diffusion equations was obtained in Watling (1992). The Brownian Riemannian bridge process was obtained by this means in Elworthy and Truman (1982). The extension to more general Riemannian manifolds was obtained in Elworthy (1988), Ndumu (1986,1991). The same methods have been applied to travelling waves for nonlinear reaction diffusion equations in Elworthy, Truman and Zhao (1994).