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Short-Time Geometry of Random Heat Kernels
Richard B. Sowers, University of Illinois, Urbana
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Memoirs of the American Mathematical Society
1998; 130 pp; softcover
Volume: 132
ISBN-10: 0-8218-0649-1
ISBN-13: 978-0-8218-0649-4
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 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.