Electronic ISBN:  9781470402181 
Product Code:  MEMO/132/629.E 
130 pp 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $30.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 132; 1998MSC: 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 LaplaceBeltrami 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 shorttime 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 nonnegligible 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.
ReadershipGraduate 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 MinakshisundaramPleijel coefficients

5. Error estimates, proof of the main theorem, and extensions


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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 LaplaceBeltrami 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 shorttime 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 nonnegligible 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.
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 MinakshisundaramPleijel coefficients

5. Error estimates, proof of the main theorem, and extensions