Heat content and inradius for regions with a Brownian boundary

Abstract

In this paper we consider $\beta [0;s]$, Brownian motion of time length $s>0$, in $m$-dimensional Euclidean space ${\mathbb{R}}^m$ and on the $m$-dimensional torus ${\mathbb{T}}^m$. We compute the expectation of (i) the heat content at time $t$ of ${\mathbb{R}}^m\setminus \beta [0;s]$ for fixed $s$ and $m=2,3$ in the limit $t\downarrow 0$, when $\beta [0;s]$ is kept at temperature 1 for all $t>0$ and ${\mathbb{R}}^m\setminus \beta [0;s]$ has initial temperature 0, and (ii) the inradius of ${\mathbb{R}}^m\setminus \beta [0;s]$ for $m=2,3,\cdots$ in the limit $s\to \infty$.

arXiv comment13 pages