Systems of nonlinear backward and forward Kolmogorov equations: generalized solutions

A probabilistic approach to construction of the solution to the Cauchy problem for systems of nonlinear parabolic equations is developed. The systems under consideration can be subdivided into two classes: the systems of the first class can be interpreted, after a simple transformation, as systems of nonlinear backward Kolmogorov equations, and the systems of the second class as systems of nonlinear forward Kolmogorov equations. By choosing an appropriate interpretation, one can construct a stochastic model in terms of a stochastic equation with coefficients depending on the solution of the Cauchy problem under consideration and the closing relation corresponding to the probabilistic representation of this solution.