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

TL;DR

This paper investigates the small-time asymptotic behavior of the heat kernel associated with a stochastically perturbed geodesic flow on a Riemannian manifold, extending WKB methods to degenerate Hamiltonians.

Contribution

It introduces a novel extension of WKB-type methods to analyze degenerate Hamiltonians in the context of stochastic geodesic flows, providing uniform bounds and asymptotic estimates.

Findings

Derived uniform bounds for degenerate Hamiltonian boundary value problems

Established two-sided estimates for the heat kernel and trace

Extended asymptotic analysis techniques to stochastic geodesic flows

Abstract

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.