  We analyze 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 WKB-type
methods to a particular case of a degenerate Hamiltonian. We derive uniform
bounds for the solution of the degenerate Hamiltonian boundary value problem
for small time. From this equivalence of solutions of the Hamiltonian equations
and the corresponding Hamilton Jacobi equation follows. The results are
exploited to derive two sided estimates and multiplicative asymptotics for the
heat kernel and the trace.
