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Brownian Motion Applied To Partial Differential Equations, Steven M. Mckay
Brownian Motion Applied To Partial Differential Equations, Steven M. Mckay
All Graduate Theses and Dissertations, Spring 1920 to Summer 2023
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 …