Brownian motion and thermal capacity

TL;DR

This paper derives an explicit formula for the maximum Hausdorff dimension of the intersection between Brownian motion images and a set, linking it to thermal capacity and Hausdorff dimension of product sets.

Contribution

It provides a new explicit formula for the Hausdorff dimension of Brownian motion intersections, connecting it to thermal capacity and space-time Hausdorff dimensions.

Findings

Explicit formula for Hausdorff dimension of intersections.

Connection between thermal capacity and intersection dimension.

Description of the formula in terms of space-time Hausdorff dimension for d≥2.

Abstract

Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset(0,\infty)$ and $F\subset \mathbf {R}^d$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when $d\ge2$, our formula can be described in terms of the Hausdorff dimension of $E\times F$, where $E\times F$ is viewed as a subspace of space time.