Stochastic Burgers' Equations and Their Semi-Classical Expansions

Abstract

In this paper we use the Hopf-Cole logarithmic transformation and the stochastic Hamilton Jacobi theory to study stochastic heat equations and Burgers' equations. Before the caustics, the stochastic inviscid Burgers' equation gives the first term of our semi-classical expansion, i.e. the inviscid limit of the viscous stochastic Burgers' equation. In order to push our results beyond the inviscid limit, we construct solutions for iterated (stochastic) Hamilton Jacobi continuity equations. Then we give the semi-classical asymptotic expansions for stochastic heat equations and Burgers' equations by using Nelson's stochastic mechanical processes with drifts given by the solution of the iterated (stochastic) Hamilton Jacobi continuity equations. The explicit formula for the remainder term is given by a path integral.