The Resource Random walk and the heat equation, Gregory F. Lawler
Random walk and the heat equation, Gregory F. Lawler
Resource Information
The item Random walk and the heat equation, Gregory F. Lawler represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of San Diego Libraries.This item is available to borrow from 1 library branch.
Resource Information
The item Random walk and the heat equation, Gregory F. Lawler represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of San Diego Libraries.
This item is available to borrow from 1 library branch.
 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
 Language
 eng
 Extent
 ix, 156 p.
 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
 Isbn
 9780821848296
 Label
 Random walk and the heat equation
 Title
 Random walk and the heat equation
 Statement of responsibility
 Gregory F. Lawler
 Subject

 Measure and integration  Classical measure theory  Fractals
 Partial differential equations  Parabolic equations and systems  Heat equation
 Probability theory and stochastic processes  Instructional exposition (textbooks, tutorial papers, etc.)
 Probability theory and stochastic processes  Markov processes  Brownian motion
 Probability theory and stochastic processes  Stochastic processes  Martingales with discrete parameter
 Probability theory and stochastic processes  Stochastic processes  Sums of independent random variables; random walks
 Random walks (Mathematics)
 Heat equation
 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
 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
 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
 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
Subject
 Measure and integration  Classical measure theory  Fractals
 Partial differential equations  Parabolic equations and systems  Heat equation
 Probability theory and stochastic processes  Instructional exposition (textbooks, tutorial papers, etc.)
 Probability theory and stochastic processes  Markov processes  Brownian motion
 Probability theory and stochastic processes  Stochastic processes  Martingales with discrete parameter
 Probability theory and stochastic processes  Stochastic processes  Sums of independent random variables; random walks
 Random walks (Mathematics)
 Heat equation
Member of
Library Links
Embed
Settings
Select options that apply then copy and paste the RDF/HTML data fragment to include in your application
Embed this data in a secure (HTTPS) page:
Layout options:
Include data citation:
<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.sandiego.edu/portal/RandomwalkandtheheatequationGregoryF./46iSwV5ltBA/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.sandiego.edu/portal/RandomwalkandtheheatequationGregoryF./46iSwV5ltBA/">Random walk and the heat equation, Gregory F. Lawler</a></span>  <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.sandiego.edu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="http://link.sandiego.edu/">University of San Diego Libraries</a></span></span></span></span></div>
Note: Adjust the width and height settings defined in the RDF/HTML code fragment to best match your requirements
Preview
Cite Data  Experimental
Data Citation of the Item Random walk and the heat equation, Gregory F. Lawler
Copy and paste the following RDF/HTML data fragment to cite this resource
<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.sandiego.edu/portal/RandomwalkandtheheatequationGregoryF./46iSwV5ltBA/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.sandiego.edu/portal/RandomwalkandtheheatequationGregoryF./46iSwV5ltBA/">Random walk and the heat equation, Gregory F. Lawler</a></span>  <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.sandiego.edu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="http://link.sandiego.edu/">University of San Diego Libraries</a></span></span></span></span></div>