Skip to main content

LUP Student Papers

LUND UNIVERSITY LIBRARIES

Brownian Motion and the Dirichlet Problem

Palets, Anton LU (2021) In Bachelor's Theses in Mathematical Sciences MATK11 20211
Mathematics (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 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.
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:
author
Palets, Anton LU
supervisor
organization
course
MATK11 20211
year
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
LUNFMA-4120-2021
ISSN
1654-6229
other publication id
2021:K23
language
English
id
9061187
date added to LUP
2021-10-14 10:53:41
date last changed
2021-10-14 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 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.}},
  author       = {{Palets, Anton}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Brownian Motion and the Dirichlet Problem}},
  year         = {{2021}},
}