Probabilistic approach for systems of second order quasi-linear parabolic PDEs

Abstract

Using the stochastic representation for second order parabolic equations, we prove the existence of local smooth solutions in Sobolev spaces for a class of second order quasi-linear parabolic partial differential equations (possibly degenerate) with smooth coefficients. As a simple application, the rate of convergence for vanishing viscosity is proved to be O(νt). Moreover, using Bismutʼs formula, we also obtain a global existence result for non-degenerate semi-linear parabolic equations. In particular, multi-dimensional Burgers equations are covered.