Harmonic functions on Walsh’s Brownian motion |
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Authors: | Patrick J Fitzsimmons Kristin E Kuter |
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Institution: | 1. Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA;2. Department of Mathematics, Saint Mary’s College, Notre Dame, IN 46556, USA |
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Abstract: | We examine a variation of two-dimensional Brownian motion introduced by Walsh that can be described as Brownian motion on the spokes of a (rimless) bicycle wheel. We construct the process by randomly assigning angles to excursions of reflecting Brownian motion. Hence, Walsh’s Brownian motion behaves like one-dimensional Brownian motion away from the origin, but differently at the origin as it is immediately sent off in random directions. Given the similarity, we characterize harmonic functions as linear functions on the rays satisfying a slope-averaging property. We also classify superharmonic functions as concave functions on the rays satisfying extra conditions. |
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Keywords: | 60G99 31A05 |
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