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

TL;DR

This paper rigorously derives the short-time asymptotics of the Dirichlet partition function for diffusion processes in tessellated Euclidean domains using path integrals and the image method, incorporating group theory.

Contribution

It introduces a group theoretic approach to derive the short-time asymptotics of the Dirichlet partition function for tessellated domains, filling a gap in the literature.

Findings

Derived the $t ightarrow 0^+$ asymptotics of $Z_{ ext{Ω}}(t)$

Applied the image method with finite reflection groups

Provided a rigorous mathematical framework for the problem

Abstract

In this paper we rigorously derive the $t\rightarrow 0^+$ 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.