Short-time Geometry of Random Heat Kernals

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.