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The Resource Random walk and the heat equation, Gregory F. Lawler

Random walk and the heat equation, Gregory F. Lawler

Label
Random walk and the heat equation
Title
Random walk and the heat equation
Statement of responsibility
Gregory F. Lawler
Creator
Subject
Language
eng
Summary
"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 equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective." "The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set." "The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas."--BOOK JACKET
Member of
Cataloging source
DLC
http://library.link/vocab/creatorDate
1955-
http://library.link/vocab/creatorName
Lawler, Gregory F.
Illustrations
illustrations
Index
no index present
LC call number
QA274.73
LC item number
.L39 2010
Literary form
non fiction
Nature of contents
bibliography
Series statement
Student mathematical library
Series volume
v. 55
http://library.link/vocab/subjectName
  • Random walks (Mathematics)
  • Heat equation
  • Probability theory and stochastic processes
  • Probability theory and stochastic processes
  • Probability theory and stochastic processes
  • Probability theory and stochastic processes
  • Partial differential equations
  • Measure and integration
Label
Random walk and the heat equation, Gregory F. Lawler
Instantiates
Publication
Bibliography note
Includes bibliographical references
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
Control code
656158762
Dimensions
22 cm
Extent
ix, 156 p.
Isbn
9780821848296
Isbn Type
(alk. paper)
Lccn
2010031593
Other physical details
ill.
System control number
(OCoLC)656158762
Label
Random walk and the heat equation, Gregory F. Lawler
Publication
Bibliography note
Includes bibliographical references
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
Control code
656158762
Dimensions
22 cm
Extent
ix, 156 p.
Isbn
9780821848296
Isbn Type
(alk. paper)
Lccn
2010031593
Other physical details
ill.
System control number
(OCoLC)656158762

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