The short-time limit of the Dirichlet partition function and the image method

Abstract

In this paper we rigorously derive the $t\to 0^{+}$ asymptotics of the free partition function $Z_{\Omega }(t)$ for a diffusion process on tessellations of the d-dimensional Euclidean space ${\mathbb{E}}^d,d=1,2,3$ with an absorbing boundary. Utilising the path integral approach and the method of images for domains which are compatible with finite reflection subgroups of the orthogonal group ${\mathbb{O}}_d$, we solve this problem following a group theoretic method which was lacking from the literature.

21 pages, 7 figures, added references, corrected typos and new material