标题： 调和测度的豪斯多夫维数 显示英文标题

标题： Hausdorff dimension of caloric measure

摘要： 我们从几何测度论的角度研究一般区域中的热量测度$\omega$，这些区域位于$\mathbb{R}^{n+1} = \mathbb{R}^n\times\mathbb{R}$（空间$\times$时间）中。 一方面，我们给出了 Taylor 和 Watson（1985）定理的一个结论的直接证明，即$\omega$的下抛物 Hausdorff 维数至少为$n$和$\omega \ll \mathcal{H}^n$。 另一方面，我们证明了$\omega$的上抛物 Hausdorff 维数最多为$n+2-\beta_n$，其中$\beta_n > 0$仅依赖于$n$。 调和测度的类似界限最初由 Nevanlinna（1934）和 Bourgain（1987）证明。 从直觉上讲，我们表明，在一个立方体内所需的障碍物密度，以使得从立方体外部开始的布朗运动不太可能在立方体中心附近退出区域，必须根据环境维度来选择。 在证明过程中，我们给出了Bourgain交替的热测度类似物：对于任何常数$0 < \epsilon \ll_n \delta < 1/2$和闭集$E \subset \mathbb{R}^{n+1}$，要么(i)对于$F$中的每个极点，$E \cap Q$在$Q \setminus E$中具有相对较大的热测度，要么(ii)对于每个$n < \rho \leq n+2$，$E \cap Q_*$具有相对较小的$\rho$维抛物Hausdorff内容，其中$Q$是一个立方体，$F$是$Q$的一个子立方体，位于顶时间面的中心对齐，而$Q_*$是$Q$的一个子立方体，它靠近但与$F$在时间上分离：$$Q = (-1/2,1/2)^n \times (-1,0), \quad F = [-1/2+\delta,1/2-\delta]^n\times[-\epsilon^2,0),$$ $$\text{and}\quad Q_* = [-1/2+\delta,1/2-\delta]^n\times[-3\epsilon^2,-2\epsilon^2].$$ 此外，我们提供了热测度的强马尔可夫性质的一种版本。 显示英文摘要

摘要： 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.