Uniform asymptotic bounds for the heat kernel and the trace of a stochastic geodesic flow

Albeverio S, Hilbert A, Kolokoltsov V. Uniform asymptotic bounds for the heat kernel and the trace of a stochastic geodesic flow. Stochastics An International Journal of Probability and Stochastic Processes. 2012;84(2-3):315-333.We analyse the asymptotic behaviour of the heat kernel defined by a stochastically perturbed geodesic flow on the cotangent bundle of a Riemannian manifold for small time and small diffusion parameter. This extends Wentzel-Kramers-Brillouin-type methods to a particular case of a degenerate Hamiltonian. We give uniform bounds for the solution of the degenerate Hamiltonian boundary value problem for small time. The results are exploited to derive two-sided estimates and multiplicative asymptotics for the heat kernel semigroup and its trace