Random Obstacle Problems

Studying the fine properties of solutions to Stochastic (Partial) Differential Equations with reflection at a boundary, this book begins with a discussion of classical one-dimensional diffusions as the reflecting Brownian motion, devoting a chapter to Bessel processes, and moves on to function-valued solutions to SPDEs. Inspired by the classical stochastic calculus for diffusions, which is unfortunately still unavailable in infinite dimensions, it uses integration by parts formulae on convex sets of paths in order to describe the behaviour of the solutions at the boundary and the contact set between the solution and the obstacle. The text may serve as an introduction to space-time white noise, SPDEs and monotone gradient systems. Numerous open research problems in both classical and new topics are proposed.

Provides a self-contained presentation in a clear and pedagogical style Includes a special chapter on Bessel processes with detailed discussions of results scattered across the literature Offers an original point of view on a booming subject (SPDEs)

Contenu
1 Introduction.- 2 The reflecting Brownian motion.- 3 Bessel processes.- 4 The stochastic heat equation.- 5 Obstacle problems.- 6 Integration by Parts Formulae.- 7 The contact set.- References.