Short-Time Asymptotics of the Heat Kernel on a Concave Bboundary

Abstract

A probabilistic method is used to study short-time asymptotic behavior of heat kernel in the exterior of an insulated smooth convex body. The expansion of the heat kernel $p(t,a,b)$ when both a and b are on the boundary is obtained by reducing the problem to the computation of a Wiener functional on a Brownian bridge. The leading terms of $\log p(t,a,b)$ are proved to be \[ - \frac{{\rho ^2 }} {{2t}} - \frac{{\mu _1 \rho ^{{1 / 3}} }} {{t^{{1 / 3}} }}\int_0^\rho {N(s)^{{2 / 3}} ds - \left( {\frac{d} {2} + \frac{1} {6}} \right)\log t + C_0 + o(1)} \] where $\rho $ is the distance between a and b, $C_0 $ is the normal curvature of the geodesic joining a and b, and $C_0 $ is an explicitly identified constant.