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.

Given a self-adjoint involution J on a Hilbert space H, we consider a
J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint
operator commuting with J and V a bounded J-self-adjoint operator
anti-commuting with J. We establish optimal estimates on the position of the
spectrum of L with respect to the spectrum of A and we obtain norm bounds on
the operator angles between maximal uniformly definite reducing subspaces of
the unperturbed operator A and the perturbed operator L.