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 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 et al. (Ann. Probab. ’15).

中文翻译：

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梳状区域和图形中的异常扩散

在本文中，我们研究了梳形平面域和梳形图中布朗运动的渐近行为。当牙齿之间的间距和牙齿宽度以相同的速率消失时，我们显示出收敛到极限过程。极限过程表现出异常的扩散行为，可以描述为布朗运动随独立粘性布朗运动的本地时间而改变。在二维设置中，主要技术步骤是对诺伊曼问题的振动估计，我们在这里使用概率论证进行证明。在一维设置中，我们既提供了直接的SDE证明，又提供了Ben Arous等人中使用捕获的布朗运动框架的证明。（Ann.Probab.'15）。