Probabilistic representations of classical and generalized Cauchy problem solutions for quasilinear systems of parabolic equations

We are going to discuss probabilistic approaches to solve the Cauchy problem for nondiagonal systems of nonlinear parabolic equations. Actually, we consider both systems diagonal in higher order terms which arise for example in connection with second order conservation laws and nondiagonal in higher order terms systems called sometimes parabolic systems with cross-diffusion. These systems arise in a number of biological problems for example in population dynamics or as a model for chemotaxis. To construct a probabilistic representation of a classical or generalized solution to the Cauchy problem for these systems, we consider the associated SDE systems with coefficients depending on distributions of their own solutions and investigate the solvability of these systems. Finally, we present a solution of the Cauchy problem for a PDE system under consideration as an average of the Cauchy data over trajectories of the corresponding SDE solution.