Stochastic Models for Nonlinear Cross-Diffusion Systems

Under a priori assumptions concerning existence and uniqueness of the Cauchy problem solution for a system of quasilinear parabolic equations with cross-diffusion, we treat the PDE system as an analogue of systems of forward Kolmogorov equations for some unknown stochastic processes and derive expressions for their generators. This allows to construct a stochastic representation of the required solution. We prove that introducing stochastic test function we can check that the stochastic system gives rise to the required generalized solution of the original PDE system. Next, we derive a closed stochastic system which can be treated as a stochastic counterpart of the Cauchy problem for a parabolic system with cross-diffusion.