We consider the asymptotic behaviour for small time and small parameter of the heat kernel on the cotangent bundle of a Riemannian manifold defined by a stochastically perturbed geodesic flow generalizing WKB‐type methods to a diffusion corresponding to a particular regular degenerate Hamiltonian. For small time equivalence of the dynamics for the degenerate Hamiltonian system and for the system given by the corresponding Hamilton‐Jacobi equation is used.