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Anceschi, Francesca, (2021)  - Sull'equazione di Kolmogorov: teoria della regolaritą ed applicazioni  - , Tesi di dottorato - (, , Universitą degli studi di Modena e Reggio Emilia ) - pagg. -

Abstract: The Kolmogorov equation was firstly introduced in 1934 as a fundamental ingredient of a kinetic model for the study of the density of a system of N particles of gas in the phase space. Kolmogorov pointed out that, although the dimension of the phase space is 2N and the diffusion term acts on the velocity variable, whose dimension is N, the differential operator is strongly degenerate. Nevertheless, Kolmogorov exhibited the explicit expression of the fundamental solution of the operator and pointed out that it is a smooth function, in fact proving that the operator is hypoelliptic. Throughout this work, we are mainly concerned with degenerate Kolmogorov equations in divergence form, for which the regularity theory for classical solutions had widely been developed during the years. Chapter 1 of this work is devoted to a survey of results on the classical regularity theory for Kolmogorov operators with constant or continuous coefficients, which can nowadays be considered complete. In Chapter 2 we consider an application of the Kolmogorov equation in finance, where the Black and Scholes theory is applied to the pricing problem for Asian options. The price of the option is computed by solving a Cauchy problem, where the initial data represents the payoff of the option and the associated PDE is a Kolmogorov type equation with local Hölder continuous coefficients. The existence and uniqueness of the fundamental solution of the associated PDO are proved, alongside with a uniqueness result for the solution of the Cauchy problem, through a limiting procedure whose convergence is ensured by Schauder type estimates. Furthermore, in Chapter 3 we consider an application of the Kolmogorov equation to the kinetic theory. Specifically, we introduce a space inhomogeneous kinetic model associated to a nonlinear Kolmogorov-Fokker-Planck (KFP) operator and we investigate the classical theory for the associated Cauchy problem. The second part of my thesis is devoted to the regularity theory for weak solutions to the Kolmogorov equation with measurable coefficients, which is nowadays the main focus of the research community. It has been developed during the last decade, and the most advanced achievements in this framework have been established in the particular case of the KFP equation. In Chapter 4 we give proof of a geometric statement for the Harnack inequality for weak solutions to the KFP equation proved by Golse, Imbert, Mouhot and Vasseur in 2017, based on the concepts of Harnack chains and attainable set. As far as we are concerned with the more general Kolmogorov equation in divergence form, Chapter 5 is devoted to the extension of the Moser’s iterative procedure for weak solutions to the Kolmogorov equation under minimal integrability assumptions for the lower order coefficients in the non dilation invariant case.