Memoirs of the American Mathematical Society 1998; 130 pp; softcover Volume: 132 ISBN10: 0821806491 ISBN13: 9780821806494 List Price: US$50 Individual Members: US$30 Institutional Members: US$40 Order Code: MEMO/132/629
 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. Readership Graduate students and research mathematicians interested in partial differential equations. Table of Contents  Introduction
 Guessing the dominant asymptotics
 Initial condition and evolution of the approximate kernel
 The MinakshisundaramPleijel coefficients
 Error estimates, proof of the main theorem, and extensions
 Appendices
 Bibliography
