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On the duration of stays of Brownian motion in domains in Euclidean space. (English) Zbl 1504.31014

Summary: Let \({T_D}\) denote the first exit time of a Brownian motion from a domain \(D\) in \({\mathbb{R}^n}\). Given domains \(U,W\subseteq{\mathbb{R}^n}\) containing the origin, we investigate the cases in which we are more likely to have fast exits from \(U\) than \(W\), meaning \(\mathbf{P}({T_U}< t)> \mathbf{P}({T_W}< t)\) for \(t\) small. We show that the primary factor in the probability of fast exits from domains is the proximity of the closest regular part of the boundary to the origin. We also prove a result on the complementary question of longs stays, meaning \(\mathbf{P}({T_U}> t)> \mathbf{P}({T_W}> t)\) for \(t\) large. This result, which applies only in two dimensions, shows that the unit disk \(\mathbb{D}\) has the lowest probability of long stays amongst all Schlicht domains.

MSC:

31B15 Potentials and capacities, extremal length and related notions in higher dimensions
60J65 Brownian motion
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