Caginalp, Carey (2011) *Analytical and Numerical Results on Escape of Brownian Particles.* Undergraduate Thesis, University of Pittsburgh.

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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 "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 given 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 sufficient amount of sample paths are compared with the exact solution to the Brownian motion (the limit of zero stepsize), providing an empirical formula for the divergence. Histograms of first hitting times are also generated.

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## Details | ||||||||||||||||

Item Type: | University of Pittsburgh ETD | |||||||||||||||
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ETD Committee: |
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Title: | Analytical and Numerical Results on Escape of Brownian Particles | |||||||||||||||

Status: | Unpublished | |||||||||||||||

Abstract: | A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called "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 given 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 sufficient amount of sample paths are compared with the exact solution to the Brownian motion (the limit of zero stepsize), providing an empirical formula for the divergence. Histograms of first hitting times are also generated. | |||||||||||||||

Date: | 29 April 2011 | |||||||||||||||

Date Type: | Completion | |||||||||||||||

Defense Date: | 28 January 2011 | |||||||||||||||

Approval Date: | 29 April 2011 | |||||||||||||||

Submission Date: | 17 March 2011 | |||||||||||||||

Access Restriction: | No restriction; The work is available for access worldwide immediately. | |||||||||||||||

Patent pending: | No | |||||||||||||||

Institution: | University of Pittsburgh | |||||||||||||||

Thesis Type: | Undergraduate Thesis | |||||||||||||||

Refereed: | Yes | |||||||||||||||

Degree: | BPhil - Bachelor of Philosophy | |||||||||||||||

URN: | etd-03172011-191325 | |||||||||||||||

Uncontrolled Keywords: | Brownian motion; stochastics | |||||||||||||||

Schools and Programs: | Dietrich School of Arts and Sciences > Mathematics University Honors College | |||||||||||||||

Date Deposited: | 10 Nov 2011 14:32 | |||||||||||||||

Last Modified: | 12 Jul 2013 07:59 | |||||||||||||||

Other ID: | http://etd.library.pitt.edu/ETD/available/etd-03172011-191325/, etd-03172011-191325 |

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