Brownian Motion and its Applications to Mathematical Analysis: École d'Été de Probabilités de Saint-Flour XLIII – 2013 (Lecture Notes in Mathematics Book 2106)

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 stock 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 stochastic analysis to Neumann eigenfunctions, Neumann heat kernel and the heat equation in time-dependent domains.-- Source other than the Library of Congress Front Matter....Pages i-xii Brownian Motion....Pages 1-10 Probabilistic Proofs of Classical Theorems....Pages 11-19 Overview of the “Hot Spots” Problem....Pages 21-29 Neumann Eigenfunctions and Eigenvalues....Pages 31-39 Synchronous and Mirror Couplings....Pages 41-62 Parabolic Boundary Harnack Principle....Pages 63-75 Scaling Coupling....Pages 77-87 Nodal Lines....Pages 89-96 Neumann Heat Kernel Monotonicity....Pages 97-105 Reflected Brownian Motion in Time Dependent Domains....Pages 107-131 Back Matter....Pages 133-140