Random Obstacle Problems: ╔cole d'╔tÚ de ProbabilitÚs de Saint-Flour XLV - 2015

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Springer, Feb 27, 2017 - Mathematics - 162 pages

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.


 

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Contents

1 Introduction
1
2 The Reflecting Brownian Motion
12
3 Bessel Processes
31
4 The Stochastic Heat Equation
59
5 Obstacle Problems
87
6 Integration by Parts Formulae
109
7 The Contact Set
141
References
158
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