Skip to main content
  • Book
  • © 2014

Brownian Motion and its Applications to Mathematical Analysis

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

Authors:

  • Contains interesting examples of couplings

  • Gentle introduction to Brownian motion and analysis

  • Heuristic explanations of the main results

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2106)

Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

This is a preview of subscription content, access via your institution.

Table of contents (10 chapters)

  1. Front Matter

    Pages i-xii
  2. Brownian Motion

    • Krzysztof Burdzy
    Pages 1-10
  3. Probabilistic Proofs of Classical Theorems

    • Krzysztof Burdzy
    Pages 11-19
  4. Overview of the “Hot Spots” Problem

    • Krzysztof Burdzy
    Pages 21-29
  5. Neumann Eigenfunctions and Eigenvalues

    • Krzysztof Burdzy
    Pages 31-39
  6. Synchronous and Mirror Couplings

    • Krzysztof Burdzy
    Pages 41-62
  7. Parabolic Boundary Harnack Principle

    • Krzysztof Burdzy
    Pages 63-75
  8. Scaling Coupling

    • Krzysztof Burdzy
    Pages 77-87
  9. Nodal Lines

    • Krzysztof Burdzy
    Pages 89-96
  10. Neumann Heat Kernel Monotonicity

    • Krzysztof Burdzy
    Pages 97-105
  11. Back Matter

    Pages 133-140

About this book

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.

Keywords

  • "hot spots" conjecture
  • 60J65, 60H30, 60G17
  • Brownian motion
  • Neumann eigenfunction
  • coupling
  • heat equation
  • partial differential equations

Authors and Affiliations

  • Department of Mathematics, University of Washington, Seattle, USA

    Krzysztof Burdzy

Bibliographic Information

  • Book Title: Brownian Motion and its Applications to Mathematical Analysis

  • Book Subtitle: École d'Été de Probabilités de Saint-Flour XLIII – 2013

  • Authors: Krzysztof Burdzy

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-04394-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2014

  • Softcover ISBN: 978-3-319-04393-7Published: 20 February 2014

  • eBook ISBN: 978-3-319-04394-4Published: 07 February 2014

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XII, 137

  • Number of Illustrations: 12 b/w illustrations, 4 illustrations in colour

  • Topics: Probability Theory, Differential Equations, Potential Theory

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions