Walsh'S Brownian Motion On A Graph

• Activity Number: 181
• Type: Contributed
• Date/Time: Monday, August 1, 2011 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Kristin Jehring*+
• Companies: Saint Mary's College
• Keywords: Brownian motion ; Markov chain ; harmonic function ; graph ; reversibility

Abstract

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.