Short-time geometry of random heat kernels

We study the short-time behavior of the random heat kernel associated with the stochastic partial differential equation du = 1/2Δudt + (σ, ∇u) o dW_t, on some Riemannian manifold M. Here Δ is the Laplace-Beltrami operator, σ is some vector field on M, and ∇ is the gradient operator. Also, W is a standard Wiener process and o denotes Stratonovich integration. Our interest here is to show how classical short-time estimates of deterministic heat kernels must be corrected to account for the noise term. We find 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. The third exponential term blows up, and thus provides an interesting object of study. We find an expression for this third exponential term which involves a random translation of the index form and the equations of Jacobi fields.