梳形区域和图中的反常扩散
Anomalous diffusion in comb-shaped domains and graphs
摘要：
本文研究了梳形平面区域和梳形图中布朗运动的渐近性态。当齿间距\emph{and}和齿宽以相同的速率消失时，我们证明了收敛到一个极限过程。该极限过程表现出反常的扩散行为，可以描述为一个布朗运动随时间变化的独立粘性布朗运动的局部时。在二维设置的主要技术步骤是一个诺依曼问题的振荡估计，我们在这里证明使用概率参数。在一维的设置中，我们提供了一个直接的证明，并使用本Arous \etal（Ann.\可能吧。'15）。
英文摘要：
In this paper we study the asymptotic behavior of Brownian motion in both comb-shaped planar domains, and comb-shaped graphs. We show convergence to a limiting process when both the spacing between the teeth \emph{and} the width of the teeth vanish at the same rate. The limiting process exhibits an anomalous diffusive behavior and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. In the two dimensional setting the main technical step is an oscillation estimate for a Neumann problem, which we prove here using a probabilistic argument. In the one dimensional setting we provide both a direct SDE proof, and a proof using the trapped Brownian motion framework in Ben Arous \etal (Ann.\ Probab.\ '15).
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