Brownian Motion and the Dirichlet Problem
(2021) In Bachelor's Theses in Mathematical Sciences MATK11 20211Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
 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 zeroone 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.
 Popular Abstract
 This work deals with two mathematical concepts from seemingly disparate worlds: the Dirichlet problem and Brownian motion. The Dirichlet problem deals with very smooth functions, whereas Brownian motion is prototypically the random movement of a particle suspended in a liquid. The intuition for the Dirichlet problem comes from physics. Imagine some object with a given temperature distribution on its surface. The problem is to find a function which would tell us the temperature at any point inside the object. This work culminates in formulating this function in terms of average properties of randomly moving particles.
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/studentpapers/record/9061187
 author
 Palets, Anton ^{LU}
 supervisor

 Yacin Ameur ^{LU}
 organization
 course
 MATK11 20211
 year
 2021
 type
 M2  Bachelor Degree
 subject
 keywords
 Brownian motion, Dirichlet, Dirichlet problem, Harmonic function, Strong Markov property, Stopping time, Cone condition, Brownian motion construction, Blumenthal
 publication/series
 Bachelor's Theses in Mathematical Sciences
 report number
 LUNFMA41202021
 ISSN
 16546229
 other publication id
 2021:K23
 language
 English
 id
 9061187
 date added to LUP
 20211014 10:53:41
 date last changed
 20211014 10:53:41
@misc{9061187, 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 zeroone 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.}}, author = {{Palets, Anton}}, issn = {{16546229}}, language = {{eng}}, note = {{Student Paper}}, series = {{Bachelor's Theses in Mathematical Sciences}}, title = {{Brownian Motion and the Dirichlet Problem}}, year = {{2021}}, }