Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations

A nonlinear Fokker-Planck type ultraparabolic integro-differential equation is studied. It arises from the statistical description of the dynamical behavior of populations of infinitely many (nonlinearly coupled) random oscillators subject to ``mean-field'' interaction. A regularized parabolic equation with bounded coefficients is first considered, where a small spatial diffusion is incorporated in the model equation and the unbounded coefficients of the original equation are replaced by a special ``bounding" function. Estimates, uniform in the regularization parameters, allow passing to the limit, which identifies a classical solution to the original problem. Existence and uniqueness of classical solutions are then established in a special class of functions decaying in the velocity variable.