Random Obstacle Problems

École d'Été de Probabilités de Saint-Flour XLV - 2015

verfasst von: Lorenzo Zambotti

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

This course is about some fine properties of stochastic processes with reflection on a boundary. In the first lectures we present some interesting one-dimensional examples, the reflecting Brownian motion and the Bessel processes. However this serves mainly as a warm-up for the next chapters where we study a class of function-valued processes. Indeed, the main focus of the course is on solutions to stochastic partial differential equations with reflection on an obstacle.

Lorenzo Zambotti

Chapter 2. The Reflecting Brownian Motion

Abstract

In this chapter we study SDEs in \(\mathbb{R}_{+}:= [0,+\infty [\) with reflection at 0. We give two different approaches: the classical one based on the Skorokhod Lemma, and the penalisation method. The interest of the latter lies in its applicability, in Chap. 5 below, to stochastic partial differential equations. We discuss the link between reflection and local times and we give a formula, in Lemma 2.9 below, which will be useful later, in particular in the construction of Bessel processes in Chap. 3

Lorenzo Zambotti

Chapter 3. Bessel Processes

Abstract

In this chapter we are going to study δ-Bessel processes, namely solutions (ρ _t)_t ≥ 0 to the SDE

Lorenzo Zambotti

Chapter 4. The Stochastic Heat Equation

Abstract

This chapter is devoted to the study of white-noises, in particular the space-time white noise, and to the first SPDE of the course, the stochastic heat equation.

Lorenzo Zambotti

Chapter 5. Obstacle Problems

Abstract

In this chapter we introduce the SPDEs with reflection which are studied in detail in the following chapters. The theory at the beginning is purely deterministic: as in the Skorohod Lemma 2.1, we have a driving continuous function (t, x) ↦ w(t, x) which plays the role of an obstacle and we look for a continuous function z ≥ −w which solves a heat equation on the open set {z > −w} and is reflected on − w. Following Nualart and Pardoux [NP92] we give an existence and uniqueness result for solutions to such obstacle problems.

Lorenzo Zambotti

Chapter 6. Integration by Parts Formulae

Abstract

Integration by Parts is one of the most basic tools of analysis and, arguably, mathematics in general. In stochastic analysis it also plays an important role, for a simple reason. Let us consider the operator \(\mathcal{L}: C_{c}^{\infty }(\mathbb{R}^{d}) \rightarrow C_{c}^{\infty }(\mathbb{R}^{d})\)

Lorenzo Zambotti

Chapter 7. The Contact Set

Abstract

In this chapter we study the contact set {u = 0} for the SPDEs with reflection (or repulsion from 0) and the reflection measure η, answering the questions raised in Sect. 1.4 In particular, we show the following surprising result: for every δ ≥ 3 and every integer \(k> \frac{4} {\delta -2},\)

Lorenzo Zambotti

Backmatter

Titel: Random Obstacle Problems
verfasst von: Lorenzo Zambotti
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-52096-4
Print ISBN: 978-3-319-52095-7
DOI: https://doi.org/10.1007/978-3-319-52096-4