Hausdorff dimension of caloric measure

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.