Short-time asymptotics of the heat kernel on a concave boundary. (English) Zbl 0685.58035

Hsu, Pei

SIAM J. Math. Anal. 20, No. 5, 1109-1127 (1989).

From the author’s abstract: A probabilistic method is used to study short-time asymptotic behaviour of heat kernels 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 \[ -\frac{\rho^ 2}{2t}- \frac{\mu_ 1\rho^{1/3}}{t^{1/3}}\int^{\rho}_{0}N(s\quad)^{2/3} ds- (\frac{d}{2}+\frac{1}{6})\log t+C_ 0+O(1) \] where \(\rho\) is the distance between a and b, N(s) is the normal curvature of the geodesic joining a and b, and \(C_ 0\) is an explicitly identified constant.

Reviewer: N.Jacob

MSC:

58J65	Diffusion processes and stochastic analysis on manifolds
35K05	Heat equation

Keywords:

heat kernel; Laplace-Beltrami operator; diffusion process on; manifold; Feynman-Kac formula; Girsanov formula; Wiener functional; Brownian bridge