Short-time Geometry of Random Heat Kernals (Memoirs of the American Mathematical Society)

Short-time Geometry of Random Heat Kernals (Memoirs of the American Mathematical Society)


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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 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.


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.

Product Details

  • ISBN13: 9780821806494
  • Format: Paperback
  • ID: 9780821806494
  • ISBN10: 0821806491

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