Brownian Motion and its Applications to Mathematical Analysis
Burdzy, Krzysztof.
Brownian Motion and its Applications to Mathematical Analysis École d'Été de Probabilités de Saint-Flour XLIII – 2013 / [electronic resource] : by Krzysztof Burdzy. - XII, 137 p. 16 illus., 4 illus. in color. online resource. - École d'Été de Probabilités de Saint-Flour, 2106 0721-5363 ; . - École d'Été de Probabilités de Saint-Flour, 2106 .
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.
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.
9783319043944
10.1007/978-3-319-04394-4 doi
Distribution (Probability theory.
Differential equations, partial.
Potential theory (Mathematics).
Probability Theory and Stochastic Processes.
Partial Differential Equations.
Potential Theory.
QA273.A1-274.9 QA274-274.9
519.2
Brownian Motion and its Applications to Mathematical Analysis École d'Été de Probabilités de Saint-Flour XLIII – 2013 / [electronic resource] : by Krzysztof Burdzy. - XII, 137 p. 16 illus., 4 illus. in color. online resource. - École d'Été de Probabilités de Saint-Flour, 2106 0721-5363 ; . - École d'Été de Probabilités de Saint-Flour, 2106 .
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.
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.
9783319043944
10.1007/978-3-319-04394-4 doi
Distribution (Probability theory.
Differential equations, partial.
Potential theory (Mathematics).
Probability Theory and Stochastic Processes.
Partial Differential Equations.
Potential Theory.
QA273.A1-274.9 QA274-274.9
519.2