Hausdorff dimension of caloric measure

TL;DR

This paper investigates the geometric properties of caloric measures in Euclidean space-time domains, establishing bounds on their Hausdorff dimensions and extending classical harmonic measure results to the caloric setting.

Contribution

It provides new bounds on the parabolic Hausdorff dimension of caloric measures and introduces a caloric measure analogue of Bourgain's alternative, advancing geometric measure theory in parabolic contexts.

Findings

Lower parabolic Hausdorff dimension of caloric measure is at least n.

Upper parabolic Hausdorff dimension of caloric measure is at most n+2−β_n.

Established a caloric measure analogue of Bourgain's alternative.

Abstract

We examine caloric measures $\omega$ on general domains in $\mathbb{R}^{n+1} = \mathbb{R}^n\times\mathbb{R}$ (space $\times$ time) from the perspective of geometric measure theory. On one hand, we give a direct proof of a consequence of a theorem of Taylor and Watson (1985) that the lower parabolic Hausdorff dimension of $\omega$ is at least $n$ and $\omega \ll \mathcal{H}^n$. On the other hand, we prove that the upper parabolic Hausdorff dimension of $\omega$ is at most $n+2-\beta_n$, where $\beta_n > 0$ depends only on $n$. Analogous bounds for harmonic measures were first shown by Nevanlinna (1934) and Bourgain (1987). Heuristically, we show that the density of obstacles in a cube needed to make it unlikely that a Brownian motion started outside of the cube exits a domain near the center of the cube must be chosen according to the ambient dimension. In the course of the proof, we give a caloric measure analogue of Bourgain's alternative: for any constants $0 < \epsilon \ll_n \delta < 1/2$ and closed set $E \subset \mathbb{R}^{n+1}$, either (i) $E \cap Q$ has relatively large caloric measure in $Q \setminus E$ for every pole in $F$ or (ii) $E \cap Q_*$ has relatively small $\rho$-dimensional parabolic Hausdorff content for every $n < \rho \leq n+2$, where $Q$ is a cube, $F$ is a subcube of $Q$ aligned at the center of the top time-face, and $Q_*$ is a subcube of $Q$ that is close to, but separated backwards-in-time from $F$: $$Q = (-1/2,1/2)^n \times (-1,0), \quad F = [-1/2+\delta,1/2-\delta]^n\times[-\epsilon^2,0),$$ $$\text{and}\quad Q_* = [-1/2+\delta,1/2-\delta]^n\times[-3\epsilon^2,-2\epsilon^2].$$ Further, we supply a version of the strong Markov property for caloric measures.