Stochastic processes associated with systems of nonlinear parabolic equations

Systems of nonlinear parabolic equations arise as mathematical models in describing of various phenomena in physics and biology. We treat such systems as systems of nonlinear forward Kolmogorov equations and derive stochastic equations for Markov processes associated with the Cauchy problem for these systems. Our aim is to derive a closed stochastic system associated with the original problem and to study it. We show that different types of solutions to the original PDE problem are connected with different (nonlinear) Markov processes. As a result we construct probabilistic representations for both generalized and measure valued solutions of nonlinear PDE systems. To illustrate general results we consider the Keller-Segel system from biology and the MHD-Burgers system from magneto-hydrodynamics.