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- W4384274310 abstract "We examine caloric measures $omega$ on general domains in $mathbb{R}^{n+1} = mathbb{R}^ntimesmathbb{R}$ (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 $F$: $$Q = (-1/2,1/2)^n times (-1,0), quad F = [-1/2+delta,1/2-delta]^ntimes[-epsilon^2,0),$$ $$text{and}quad Q_* = [-1/2+delta,1/2-delta]^ntimes[-3epsilon^2,-2epsilon^2].$$ Further, we supply a version of the strong Markov property for caloric measures." @default.
- W4384274310 created "2023-07-15" @default.
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- W4384274310 date "2021-08-27" @default.
- W4384274310 modified "2023-10-18" @default.
- W4384274310 title "Hausdorff dimension of caloric measure" @default.
- W4384274310 doi "https://doi.org/10.48550/arxiv.2108.12340" @default.
- W4384274310 hasPublicationYear "2021" @default.
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