Heat content and inradius for regions with a Brownian boundary

TL;DR

This paper analyzes the heat content and inradius of regions with a Brownian boundary in Euclidean space and on the torus, providing explicit expectations in certain limits for dimensions 2 and 3.

Contribution

It computes the expected heat content and inradius for regions with a Brownian boundary, extending understanding of stochastic geometric properties in Euclidean and toroidal spaces.

Findings

Expected heat content at small times for 2D and 3D cases.

Asymptotic behavior of inradius as Brownian motion duration tends to infinity.

Results applicable to Euclidean space and torus geometries.

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 $s \rightarrow \infty$.