Short-Time Geometry of Random Heat Kernels (Memoirs of the AMS) (Memoirs of the American Mathematical Society) - Softcover

American Mathematical Society

Softcover

ISBN 10: 0821806491 ISBN 13: 9780821806494

Publisher: American Mathematical Society, 1998

Synopsis

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.

Edition details

• Publisher: American Mathematical Society
• Publication date: 1998
• Language: English
• ISBN 10: 0821806491
• ISBN 13: 9780821806494
• Binding: Paperback
• Number of pages: 130