Recent Results on the Short-Time Geometry of Random Heat Kernels

We give some recent results concerning the short-time behavior of the random heat kernel associated with the stochastic partial differential equation $du = \frac12 \Delta u dt + (\sigma,\grad u)\circ dW_t$ on some Riemannian manifold~$M$. Here $\Delta$ is the Laplace-Beltrami operator, $\sigma$ is some vector field, $\grad$ is the gradient operator, and ${}\circ dW_t$ denotes Stratonovich integration against a standard Wiener process. These results show how classical short-time asymptotics of deterministic heat kernels must be corrected to account for the random term; an exponential term must be added.

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Published 1 January 1994