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Dirichlet Problem with Brownian Motion April 10, 2010

Posted by Sarah in Uncategorized.
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I learned this in my probability class and thought it was pretty neat.

The Dirichlet problem is a standard problem in differential equations.
Given a region in the plane, its boundary , and a function on the boundary, we want to solve for such that is harmonic, and . (Harmonic just has the usual meaning that the Laplacian is zero: .)

The probabilistic approach is to take a Brownian motion starting at a point Let be the first moment in time when .Consider the function We claim that this is a solution to the Dirichlet problem.

To show this we need to determine
1. is harmonic.
2. Under some condition as

To prove 1, we use a mean value property


which just says that the integral around a circle is the value of the function at the center. This is a consequence of the strong Markov property for Brownian motion.
Expanding in a Taylor series gives us


Using the integral and noting that odd functions integrate to 0, we get

Which shows that is harmonic.

To prove 2, we use the fact that for a given point in the plane, the probability of any Brownian motion passing through that point is zero.

We claim that we have convergence under the following condition:
as if . Then we have

To do this, we use some computations with integrals and inequalities; I’ve been trying to put this up but WordPress and LaTeX hate me and are giving me all kinds of errors, so this part will sadly have to be sans proof.

I like this little thing because it illustrates the relationship between harmonic functions and diffusion.

(Note: this is my first time using LaTeX in WordPress! I’m so happy. I use Sitmo.

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