Heat Content and Inradius for Regions with a Brownian Boundary

Abstract

Listen

Translate article

In this paper we consider β[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 ${\mathbb R}^m \backslash \beta [0,s]$ for fixed s and m = 2, 3 in the limit t ↓ 0, when β[0, s] is kept at temperature 1 for all t > 0 and ${\mathbb R}^m \backslash \beta [0,s]$ has initial temperature 0, and (ii) the inradius of ${\mathbb T}^m \backslash \beta [0,s]$ for m = 2, 3, ⋯ in the limit s → ∞.