Stochastic interpretation of quasilinear parabolic systems

In this talk we discuss connections between the Markov process theory and the theory of generalized solutions of the Cauchy problem for systems of nonlinear parabolic equations. These systems naturally arise as mathematical models of conservation laws in various physical and biological problems as well as models of some problems in economics and financial mathematics. Probabilistic representations of generalized solutions of the Cauchy problem for systems of semilinear and quasilinear parabolic equations have been constructed in our earlier papers in terms of certain diffusion processes and their operator multiplicative functionals. In that case the main technical tools were both classical and generalized Ito formulas as well as forward and backward stochastic flows generated by solutions of corresponding SDEs. For parabolic systems with non diagonal principal part (called systems with cross-diffusion) we construct the corresponding probabilistic representations via interpretation of these systems as systems for densities of interacting Markov processes. Generators of these processes are defined by the integral identities used to introduce the notion of a generalized solution of a parabolic system. Besides we define stochastic test functions and choose special multiplicative functionals in order to construct the required probabilistic representations of generalized solutions to the Cauchy problem under consideration.