Short-time geometry of random heat kernels Book

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.