Brownian Motion and Partial Differential Equations

Abstract

In this chapter, we use the results of the preceding two chapters to discuss connections between Brownian motion and partial differential equations. After a brief discussion of the heat equation, we focus on the Laplace equation Δ u = 0 and on the relations between Brownian motion and harmonic functions on a domain of $\mathbb{R}^{d}$. In particular, we give the probabilistic solution of the classical Dirichlet problem in a bounded domain whose boundary satisfies the exterior cone condition. In the case where the domain is a ball, the solution is made explicit by the Poisson kernel, which corresponds to the density of the exit distribution of the ball for Brownian motion. We then discuss recurrence and transience of d-dimensional Brownian motion, and we establish the conformal invariance of planar Brownian motion as a simple corollary of the results of Chap. 5 An important application is the so-called skew-product decomposition of planar Brownian motion, which we use to derive several asymptotic laws, including the celebrated Spitzer theorem on Brownian windings.