The Feynman-Kac formula and applications to PDEs

Gallucci, Anna (2014) The Feynman-Kac formula and applications to PDEs. [Laurea magistrale], Università di Bologna, Corso di Studio in Matematica [LM-DM270], Documento ad accesso riservato.
Documenti full-text disponibili:
[img] Documento PDF
Full-text accessibile solo agli utenti istituzionali dell'Ateneo

Download (479kB) | Contatta l'autore


The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Tipologia del documento
Tesi di laurea (Laurea magistrale)
Autore della tesi
Gallucci, Anna
Relatore della tesi
Corso di studio
Curriculum A: Generale e applicativo
Ordinamento Cds
Parole chiave
Feynman-Kac formula Markov semigroups principal eigenvalue
Data di discussione della Tesi
12 Dicembre 2014

Altri metadati

Statistica sui download

Gestione del documento: Visualizza il documento