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abstract: We examine caloric measures $$\omega$$ on general domains in $$\RR^{n+1}=\RR^n\times\RR$$ (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 $$\omega\ll\Haus^n$$. 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 \emph{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\RR^{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$$: \begin{gather*} Q=(-1/2,1/2)^n\times (-1,0),\quad F=[-1/2+\delta,1/2-\delta]^n\times[-\epsilon^2,0),\\[2pt] \text{and }Q_*=[-1/2+\delta,1/2-\delta]^n\times[-3\epsilon^2,-2\epsilon^2]. \end{gather*} Further, we supply a version of the strong Markov property for caloric measures. 

Abstract For every$$n\geq 2$$, Bourgain’s constant$$b_n$$is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most$$n-b_n$$for every domain in$$\mathbb {R}^n$$on which harmonic measure is defined. Jones and Wolff (1988,Acta Mathematica161, 131–144) proved that$$b_2=1$$. When$$n\geq 3$$, Bourgain (1987,Inventiones Mathematicae87, 477–483) proved that$$b_n>0$$and Wolff (1995,Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing$$b_n<1$$. Refining Bourgain’s original outline, we prove that$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$for all$$n\geq 3$$, where$$c>0$$is a constant that is independent ofn. We further estimate$$b_3\geq 1\times 10^{-15}$$and$$b_4\geq 2\times 10^{-26}$$.