Stochastic models for forward systems of nonlinear parabolic equations

Abstract

Stochastic differential equations give a constructive way to obtain a large number of stochastic processes. They can be used as well to simulate these processes numerically. As a by-product we can construct numerically solutions of the Cauchy problem for some parabolic equations and systems. In this paper we derive systems of stochastic equations for stochastic processes underlying systems of coupled parabolic equations which are particular cases of cross-diffusion systems of PDEs. Stochastic processes satisfying these stochastic systems allow to obtain probabilistic representations of classical and weak solutions of the Cauchy problem for original PDE systems and can be used to simulate the PDE solutions.