Regularity results and new perspectives for degenerate Kolmogorov equations

Abstract

In this thesis, we are concerned with the regularity theory of strongly degenerate Kolmogorov equations and we also study a relativistic generalization of such equations. We divide this dissertation into three parts. In the first part, we present some results which lie in the classical regularity theory of Kolmogorov-type operators with regular coefficients. In particular, we here discuss some Schauder estimates for classical solutions to Kolmogorov equations in non-divergence form with Dini-continuous coefficients and right-hand side. Furthermore, we show new pointwise regularity results and a Taylor-type expansion up to second order with estimate of the rest in L^p norm. The second part focuses on the weak regularity theory of degenerate Kolmogorov equations with discontinuous coefficients, which is nowadays the main focus of the research community. More precisely, we present a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation with measurable coefficients, integrable lower order terms and nonzero source term. We subsequently prove the existence of a fundamental solution associated to the Kolmogorov operator, together with Gaussian lower and upper bounds. Finally, in the last part of this thesis, we address a possible generalization of the kinetic Kolmogorov-Fokker-Planck equation, which is in accordance with the theory of special relativity. In particular, we explain why the operator proposed is the suitable relativistic generalization of the Fokker-Planck operator and we describe it as a Hörmander operator which is invariant with respect to Lorentz transformations.