Date of Award:

5-1985

Document Type:

Thesis

Degree Name:

Master of Science (MS)

Department:

Mathematics and Statistics

Advisor/Chair:

Michael Brennan

Abstract

This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.

Checksum

8ac36d1569d5cb7c99fc604d1c6d20eb

Included in

Mathematics Commons

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