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Open Access Publications from the University of California

Harmonic functions on Walsh's Brownian motion

  • Author(s): Jehring, Kristin Elizabeth
  • et al.

In this dissertation 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 R² 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. Taking advantage of this similarity, we provide a characterization of harmonic functions for the process as linear functions on the rays that satisfy a slope- averaging property. We also classify superharmonic functions as concave functions on the rays satisfying some extra conditions. We then generalize the state space to consider a process on any connected, locally finite graph obtained by gluing a number of Walsh's Brownian motion processes in R² together. In this generalized situation, we also 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. We address the reversibility of the process and derive the Dirichlet form of the reversible Walsh's Brownian motion on a graph.

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