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Short-Time Geometry of Random Heat Kernels
 
Richard B. Sowers University of Illinois, Urbana, Urbana, IL
Front Cover for Short-Time Geometry of Random Heat Kernels
Available Formats:
Electronic ISBN: 978-1-4704-0218-1
Product Code: MEMO/132/629.E
130 pp 
List Price: $50.00
MAA Member Price: $45.00
AMS Member Price: $30.00
Front Cover for Short-Time Geometry of Random Heat Kernels
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  • Front Cover for Short-Time Geometry of Random Heat Kernels
  • Back Cover for Short-Time Geometry of Random Heat Kernels
Short-Time Geometry of Random Heat Kernels
Richard B. Sowers University of Illinois, Urbana, Urbana, IL
Available Formats:
Electronic ISBN:  978-1-4704-0218-1
Product Code:  MEMO/132/629.E
130 pp 
List Price: $50.00
MAA Member Price: $45.00
AMS Member Price: $30.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1321998
    MSC: Primary 35; 60; 93; Secondary 58;

    This volume studies the behavior of the random heat kernel associated with the stochastic partial differential equation \(du=\tfrac {1}{2} {\Delta}udt = (\sigma, \nabla u) \circ dW_t\), on some Riemannian manifold \(M\). Here \(\Delta\) is the Laplace-Beltrami operator, \(\sigma\) is some vector field on \(M\), and \(\nabla\) is the gradient operator. Also, \(W\) is a standard Wiener process and \(\circ\) denotes Stratonovich integration. The author gives short-time expansion of this heat kernel. He finds that the dominant exponential term is classical and depends only on the Riemannian distance function. The second exponential term is a work term and also has classical meaning. There is also a third non-negligible exponential term which blows up. The author finds an expression for this third exponential term which involves a random translation of the index form and the equations of Jacobi fields. In the process, he develops a method to approximate the heat kernel to any arbitrary degree of precision.

    Readership

    Graduate students and research mathematicians interested in partial differential equations.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Guessing the dominant asymptotics
    • 3. Initial condition and evolution of the approximate kernel
    • 4. The Minakshisundaram-Pleijel coefficients
    • 5. Error estimates, proof of the main theorem, and extensions
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Volume: 1321998
MSC: Primary 35; 60; 93; Secondary 58;

This volume studies the behavior of the random heat kernel associated with the stochastic partial differential equation \(du=\tfrac {1}{2} {\Delta}udt = (\sigma, \nabla u) \circ dW_t\), on some Riemannian manifold \(M\). Here \(\Delta\) is the Laplace-Beltrami operator, \(\sigma\) is some vector field on \(M\), and \(\nabla\) is the gradient operator. Also, \(W\) is a standard Wiener process and \(\circ\) denotes Stratonovich integration. The author gives short-time expansion of this heat kernel. He finds that the dominant exponential term is classical and depends only on the Riemannian distance function. The second exponential term is a work term and also has classical meaning. There is also a third non-negligible exponential term which blows up. The author finds an expression for this third exponential term which involves a random translation of the index form and the equations of Jacobi fields. In the process, he develops a method to approximate the heat kernel to any arbitrary degree of precision.

Readership

Graduate students and research mathematicians interested in partial differential equations.

  • Chapters
  • 1. Introduction
  • 2. Guessing the dominant asymptotics
  • 3. Initial condition and evolution of the approximate kernel
  • 4. The Minakshisundaram-Pleijel coefficients
  • 5. Error estimates, proof of the main theorem, and extensions
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