Uniform asymptotic bounds for the heat kernel and the trace of a stochastic geodesic flow

Abstract

We analyse 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 Wentzel–Kramers–Brillouin-type methods to a particular case of a degenerate Hamiltonian. We give uniform bounds for the solution of the degenerate Hamiltonian boundary value problem for small time. The results are exploited to derive two-sided estimates and multiplicative asymptotics for the heat kernel semigroup and its trace.

Keywords:

• asymptotic expansions
• heat kernel
• stochastic geodesic flow
• trace formula

AMS Subject Classification::

• 46N20
• 46N30
• 60H10

Acknowledgements

The authors would like to thank Zdisláv Brzeźniak and David Elworthy for stimulating discussions. We are indebted to the referee for his valuable suggestions and for completing the references. The second author is particularly grateful to Rémi Léandre for introducing her to the theory of small parameter estimates and gratefully acknowledges financial support by ‘Profilen matematisk modellering’ at Växjö University. The authors are deeply indebted to Paul Fischer for his help with programming in MAPLE which was essential for the computations.