Introducció a les equacions en derivades parcials estocàstiques

Abstract

[en] This work is an introduction to Stochastic partial differential equations (SPDEs). We study the stochastic calculus needed to define the equations, discuss the existence and uniqueness of solutions and, in the last place, we make use of the Malliavin calculus in order to study some properties of the law of the solutions of some SPDEs. The first chapter is devoted to the definition of the space-time white noise. Following John B. Walsh’s theory [5], we define the stochastic integral of deterministic and random functions with respect to the white noise. The second chapter is devoted to the understanding of what partial differential equations (PDEs) are, which phenomena do they model and what variations lead to the random model. We state and prove a result about the existence and uniqueness of solutions and we apply it to the particular cases of the stochastic heat equation and stochastic wave equation. Finally, in the third chapter we study the Malliavin calculus. This is a theory (initiated by Paul Malliavin and developed by David Nualart and Marta Sanz [12] [13]) that extends the classical rules of differential and integral calculus to random variables. We study the relationship between the Malliavin calculus and the law of random variables, stating and proving a result about the absolute continuity of the solutions of some SPDEs.