Description:
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
Brownian motion -- Probabilistic proofs of classical theorems -- Overview of the "hot spots" problem -- Neumann eigenfunctions and eigenvalues -- Synchronous and mirror couplings -- Parabolic boundary Harnack principle -- Scaling coupling -- Nodal lines -- Neumann heat kernel monotonicity -- Reflected Brownian motion in time dependent domains