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Abstract Details
Activity Number:
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181
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Type:
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Contributed
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Date/Time:
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Monday, August 1, 2011 : 10:30 AM to 12:20 PM
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Sponsor:
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IMS
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Abstract - #303126 |
Title:
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Walsh'S Brownian Motion On A Graph
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Author(s):
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Kristin Jehring*+
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Companies:
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Saint Mary's College
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Address:
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Department of Mathematics, Notre Dame, IN, 46556,
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Keywords:
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Brownian motion ;
Markov chain ;
harmonic function ;
graph ;
reversibility
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Abstract:
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We examine a variation of two-dimensional Brownian motion introduced in 1978 by Walsh. Walsh's Brownian motion can be described as a Brownian motion on the spokes of a (rimless) bicycle wheel. We will construct such a process by randomly assigning an angle to the excursions of a reflecting Brownian motion from 0. With this construction we see that Walsh's Brownian motion in the plane behaves like one-dimensional Brownian motion away from the origin, but at the origin behaves differently as the process is sent off in another random direction. We generalize the state space to consider a process on any connected, locally finite graph obtained by gluing a number of planar Walsh's Brownian motion processes together. In this generalized situation, we classify harmonic functions. We introduce a Markov chain associated to Walsh's Brownian motion on a graph and explore the relationship between the two processes, specifically the reversibility of the two processes. We derive formulas for the transition probabilities of the so-called embedded Markov chain and for the passage times of the Walsh's Brownian motion.
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