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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 We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/ s )-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s . At the endpoint, α = s , a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/ s )-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s -dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n , the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n .