Brownian Motion and its Applications to Mathematical Analysis - École d'Été de Probabilités de Saint-Flour XLIII - 2013

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.

Sommario

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.

Riassunto

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.

Dettagli sul prodotto

• Autori: Krzysztof Burdzy
• Editore: Springer, Berlin
• Lingue: Inglese
• Formato: Tascabile
• Pubblicazione: 16.12.2013
• EAN: 9783319043937
• ISBN: 978-3-31-904393-7
• Pagine: 137
• Dimensioni: 159 mm x 237 mm x 10 mm
• Peso: 254 g
• Illustrazioni: XII, 137 p. 16 illus., 4 illus. in color.
• Serie: Lecture Notes in Mathematics; École d'Été de Probabilités de Saint-Flour; Lecture Notes in Mathematics / École d'Été de Probabilités de Saint-Flour; Lecture Notes in Mathematics; École d'Été de Probabilités de Saint-Flour
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Teoria delle probabilità, stocastica, statistica matematica; B, Mathematische Analysis, allgemein, Mathematics and Statistics, Probability Theory and Stochastic Processes, Partial Differential Equations, Differential calculus & equations, Differential equations, Probabilities, Stochastics, Probability Theory, Potential Theory, Potential theory (Mathematics)