Random walk and the heat equation / Gregory F. Lawler.

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"The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equati...

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Bibliographic Details
Main Author: Lawler, Gregory F., 1955-
Language:English
Published: Providence, R.I. : American Mathematical Society, [2010], ©2010.
Series:Student mathematical library ; v. 55.
Subjects:
Random walks (Mathematics)
Heat equation.
Physical Description:ix, 156 pages : illustrations ; 22 cm.
Format: Book
Contents:
  • Chapter 1. Random Walk and Discrete Heat Equation
  • 1.1. Simple random walk
  • 1.2. Boundary value problems
  • 1.3. Heat equation
  • 1.4. Expected time to escape
  • 1.5. Space of harmonic functions
  • 1.6. Exercises
  • Chapter 2. Brownian Motion and the Heat Equation
  • 2.1. Brownian motion
  • 2.2. Harmonic functions
  • 2.3. Dirichlet problem
  • 2.4. Heat equation
  • 2.5. Bounded domain
  • 2.6. More on harmonic functions
  • 2.7. Constructing Brownian motion
  • 2.8. Exercises
  • Chapter 3. Martingales
  • 3.1. Examples
  • 3.2. Conditional expectation
  • 3.3. Definition of martingale
  • 3.4. Optional sampling theorem
  • 3.5. Martingale convergence theorem
  • 3.6. Uniform integrability
  • 3.7. Exercises
  • Chapter 4. Fractal Dimension
  • 4.1. Box dimension
  • 4.2. Cantor measure
  • 4.3. Hausdorff measure and dimension
  • 4.4. Exercises.