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We analyze the asymptotic behaviour of the heat kernel defined by astochastically perturbed geodesic flow on the cotangent bundle of a Riemannianmanifold for small time and small diffusion parameter. This extends WKB-typemethods to a particular case of a degenerate Hamiltonian. We derive uniformbounds for the solution of the degenerate Hamiltonian boundary value problemfor small time. From this equivalence of solutions of the Hamiltonian equationsand the corresponding Hamilton Jacobi equation follows. The results areexploited to derive two sided estimates and multiplicative asymptotics for theheat kernel and the trace.

Asymptotic Expansions for the Heat Kernel and the Trace of a Stochastic Geodesic Flow