Brownian Motion and its Applications to Mathematical Analysis: École d'Été de Probabilités de Saint-Flour XLIII - 2013

by Krzysztof Burdzy

Overview

These lecture notes provide an introduction to the applications of Brownian motion to analysis and more generally, connections between Brownian motion and analysis. Brownian motion is a well-suited model for a wide range of real random phenomena, from chaotic oscillations of microscopic objects, such as flower pollen in water, to sk market fluctuations. It is also a purely abstract mathematical tool which can be used to prove theorems in "deterministic" fields of mathematics.

The notes include a brief review of Brownian motion and a section on probabilistic proofs of classical theorems in analysis. The bulk of the notes are devoted to recent (post-1990) applications of shastic analysis to Neumann eigenfunctions, Neumann heat kernel and the heat equation in time-dependent domains.

Product Details

ISBN-13:	9783319043937
Publisher:	Springer International Publishing
Publication date:	02/08/2014
Series:	Lecture Notes in Mathematics , #2106
Edition description:	2014
Pages:	137
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

1. Brownian motion.- 2. Probabilistic proofs of classical theorems.- 3. Overview of the "hot spots" problem.- 4. Neumann eigenfunctions and eigenvalues.- 5. Synchronous and mirror couplings.- 6. Parabolic boundary Harnack principle.- 7. Scaling coupling.- 8. Nodal lines.- 9. Neumann heat kernel monotonicity.- 10. Reflected Brownian motion in time dependent domains.