Short-time geometry of random heat kernels - 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.