Brownian Motion and the Dirichlet Problem

University essay from Lunds universitet/Matematik LTH; Lunds universitet/Matematik (naturvetenskapliga fakulteten)

Author: Anton Palets; [2021]

Keywords: Brownian motion; Dirichlet; Dirichlet problem; Harmonic function; Strong Markov property; Stopping time; Cone condition; Brownian motion construction; Blumenthal; Mathematics and Statistics;

Abstract: In this Bachelor's thesis, a solution to the Dirichlet problem using Brownian motion is given. Brownian motion is constructed using Kolmogorov's existence and continuity theorems. Blumenthal's zero-one law and the strong Markov property in various formulations are proven. Using these results, a solution to the Dirichlet problem is given using Brownian motion. The cone condition which gives conditions on the domain guaranteeing existence of solution is proven.

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