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Analytical and Numerical Results for an Escape Problem

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Abstract

A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called a “window” or “gate”) in which it is absorbed. The main problems are to determine the first hitting time and spatial distribution. A closed formula for the mean first hitting time is discovered and proven for a gate of any size. Also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed. Numerical simulations of the stochastic process with finite step size and a sufficient number of sample paths are compared with the exact solution to the Brownian motion (the limit of zero step size), providing an empirical formula for the difference. Histograms of first hitting times are also generated.

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Author notes

  1. Carey Caginalp

    Present address: Applied Mathematics, Brown University, Box F, 182 George Street, Providence, RI, 02912, USA

Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Carey Caginalp & Xinfu Chen

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  1. Carey Caginalp
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  2. Xinfu Chen
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Correspondence to Carey Caginalp.

Additional information

Communicated by C. Dafermos

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Caginalp, C., Chen, X. Analytical and Numerical Results for an Escape Problem. Arch Rational Mech Anal 203, 329–342 (2012). https://doi.org/10.1007/s00205-011-0455-6

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  • DOI: https://doi.org/10.1007/s00205-011-0455-6

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