Una introducció a les equacions en derivades parcials estocàstiques

Abstract

[en] In this project we introduce stochastic partial differential equations (SPDEs), focusing on the stochastic wave equation. We start by defining the Wiener integral with respect to space-time white noise. This integral allows us to provide the notion of random field solution of an SPDE. Once we have shown the existence of random field solution of the one-dimensional stochastic linear wave equation, our main goal will be to prove results on the Hölder continuity of its sample paths when the domains are $\mathbb{R}_{+}$ and [0, $L$].

We also study the nonlinear case, which requires the use of another integral whose integrand can be random, in contrast to the Wiener integral. To define this integral, we follow Walsh’s approach [13]. We prove a theorem on existence and uniqueness of random field solution to nonlinear SPDEs and then state sufficient conditions yielding the Hölder continuity of its sample paths. Finally, we apply these results to conclude that the stochastic nonlinear wave equation has a random field solution and its sample paths are jointly Hölder continuous under some given conditions.