Brownian motion and the classical Dirichlet problem 0: Introduction

In this sequence of posts I will present a probabilistic solution of the classical Dirichlet problem using Brownian motion. I will follow the structure of the book Green, Brown and Probability and Kai-Lai Chung with some little changes and somewhat more explanation. We assume knowledge of “advanced calculus”, real variables and some probability theory. No knowledge of Dirichlet problem and Brownian motion is assumed. We hope that the posts are accessible to undergraduates.

Convention: For convenience, we will work solely on the Euclidean space {\Bbb R}^3. It turns out that the theory is much the same for all spaces {\Bbb R}^d with d \geq 3. The theory for d = 2 is quite different, as we will see. \Box

We begin by recalling the classical Dirichlet problem. Let D be a domain in {\Bbb R}^3. By definition, this means that D is a non-empty connected open set in {\Bbb R}^3. A real-valued function f defined on D is said to be harmonic on D if u if the second derivatives of u are continuous on D (this we denote by writing u \in C^2(D), and

\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = \sum_{i = 1}^3 \frac{\partial^2 u}{\partial x_i^2} = 0

on D. The operator \Delta is called the Laplacian.

Classical Dirichlet Problem: Let f be a continuous function on \partial D, the boundary of D. Find a function u with the following properties:

i) u is continuous on \overline{D}.

ii) u is harmonic on D.

iii) u = f on \partial D.

The function u is said to be a solution of the Dirichlet problem. \Box

An example in electrostatics. To put things into perspective, let us describe a situation where this equation arises. Recall that in electrostatics, the force between two point charges Q and q equals

F = k \frac{Qq}{r^2},

where r is the distance between the charges and k is some constants depending on the unit. (This is called Coulomb’s Law.) The charge Q (say at x_0) creates a potential in space, given by

U(x) = k \frac{Q}{|x - x_0|}.

Suppose now that we have a distribution of charges on the surface of some conductor, two parallel metal plates. The distribution can be described in terms of a density function \rho(y), where y ranges over the conductor. The charge creates a potential

U(x) = \int k \frac{\rho(y)}{|x - y|} d\sigma(y).

Here the integration is over the conductor (two surfaces in this case). Here is the key fact:

The potential function U is harmonic outside the conductor.

To understand this at least intuitively, you should check that u_{x_0}(x) = \frac{1}{|x - x_0|} is harmonic outside the singularity x_0. Then integrating still gives a harmonic function (Exercise: Make this precise under appropriate conditions). Suppose that the potential on the plates are fixed. Then the problem of determining the potential between the plates can be modeled as a Dirichlet problem. \Box

Now we have a well posed mathematical problem, namely the Dirichlet problem. (I will omit the word “classical” from now on.) In dealing with a boundary value problem like this, we would like to ask the following questions:

1) Given the domain D and a boundary continuous function f, can we always solve the Dirichlet problem?

2) If a solution exists, is it unique?

3) Are there ways, or formulas, for computing solutions?

You may already know the answers from a previous course on partial differential equations or private reading. The purpose in this notes is to obtain and explain these results from the perspective of Brownian motion. There are very deep connection behind these two seemingly unrelated theories. We prefer not to spell out the relationship at the moment…

This ends the introduction. In the next post we will review Green’s identity and define Green’s function. After that we will give a hint of why Brownian motion is relevant, if at all.

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