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

• Introduction
• Guessing the dominant asymptotics
• Initial condition and evolution of the approximate kernel
• The Minakshisundaram-Pleijel coefficients
• Error estimates, proof of the main theorem, and extensions
• Appendices
• Bibliography