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 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:
http://lup.lub.lu.se/student-papers/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
- 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}}, }