Brownian Motion and its Applications to Mathematical Analysis

École d'Été de Probabilités de Saint-Flour XLIII – 2013

Nonfiction, Science & Nature, Mathematics, Differential Equations, Statistics
Cover of the book Brownian Motion and its Applications to Mathematical Analysis by Krzysztof Burdzy, Springer International Publishing
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Author: Krzysztof Burdzy ISBN: 9783319043944
Publisher: Springer International Publishing Publication: February 7, 2014
Imprint: Springer Language: English
Author: Krzysztof Burdzy
ISBN: 9783319043944
Publisher: Springer International Publishing
Publication: February 7, 2014
Imprint: Springer
Language: English

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.

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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.

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