Brownian Motion and its Applications to Mathematical Analysis
École d'Été de Probabilités de Saint-Flour XLIII – 2013
Burdzy, Krzysztof.
creator
author.
SpringerLink (Online service)
text
gw
2014
monographic
eng
access
XII, 137 p. 16 illus., 4 illus. in color. online resource.
These lecture notes provide an introduction to the applications of Brownian motion to analysis and, more generally, connections between Brownian motion and analysis. Brownian motion is a well-suited model for a wide range of real random phenomena, from chaotic oscillations of microscopic objects, such as flower pollen in water, to stock market fluctuations. It is also a purely abstract mathematical tool which can be used to prove theorems in "deterministic" fields of mathematics. The notes include a brief review of Brownian motion and a section on probabilistic proofs of classical theorems in analysis. The bulk of the notes are devoted to recent (post-1990) applications of stochastic analysis to Neumann eigenfunctions, Neumann heat kernel and the heat equation in time-dependent domains.
1. Brownian motion -- 2. Probabilistic proofs of classical theorems -- 3. Overview of the "hot spots" problem -- 4. Neumann eigenfunctions and eigenvalues -- 5. Synchronous and mirror couplings -- 6. Parabolic boundary Harnack principle -- 7. Scaling coupling -- 8. Nodal lines -- 9. Neumann heat kernel monotonicity -- 10. Reflected Brownian motion in time dependent domains.
by Krzysztof Burdzy.
Mathematics
Differential equations, partial
Potential theory (Mathematics)
Distribution (Probability theory)
Mathematics
Probability Theory and Stochastic Processes
Partial Differential Equations
Potential Theory
QA273.A1-274.9
QA274-274.9
519.2
Springer eBooks
Lecture Notes in Mathematics, 2106
9783319043944
http://dx.doi.org/10.1007/978-3-319-04394-4
http://dx.doi.org/10.1007/978-3-319-04394-4
140207
20150803155057.0
978-3-319-04394-4