On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form (∂_t + X · ∇_Y)u = ∇_X · (A(∇_Xu, X, Y, t)). The function A = A(ξ, X, Y, t): ℝ^m × ℝ^m × ℝ^m × ℝ → ℝ^m is assumed to be continuous with respect to ξ, and measurable with respect to X, Y and t. A = A(ξ, X, Y, t) is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded X, Y and t dependent domains.