Probabilistic Background of Viscous Conservation and Balance Laws

The aim of this paper is to construct stochastic processes allowing to obtain probabilistic representations of classical, weak or viscosity solutions of the forward Cauchy problem for several types of systems of nonlinear PDEs arising as viscous conservation and balance laws in various applications. The required stochastic processes are constructed as solutions of corresponding stochastic differential equations (SDEs) both forward and backward in time. Due to non-linearity of PDE systems under consideration additional relations must be added to the SDEs in order to obtain closed systems that can be studied independently. These relations are proved to generate probabilistic representations of the required solutions of the Cauchy problem for the original nonlinear PDE systems. Probabilistic representations are used to develop new numerical algorithms for approximation of classical and viscosity solutions to nonlinear PDEs.