We consider weak solutions of second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in the formwhere A is an uniformly positive symmetric matrix with bounded measurable coefficients, f and the components of the vector b are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a strong maximum principle.

We prove L ∞ loc estimates for positive solutions to the following degenerate second order partial differential equation of Kolmogorov type with measurable coefficients of the formis an uniformly positive symmetric matrix with bounded measurable coefficients, (b ij ) is a constant matrix. We apply the Moser's iteration method to prove the local boundedness of the solution u under minimal integrability assumption on the coefficients.

The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation L u = f with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a function space W, suitable for the study of weak solutions to L u = f , that allows us to prove a weak Poincaré inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Krȗzkov (1963). Then this functional inequality is combined with a classical covering argument (Ink-Spots Theorem) that we extend for the first time to the case of ultraparabolic equations.

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We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.