Random walks on almost connected locally compact groups: Boundary and convergence |
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Authors: | Wojciech Jaworski |
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Institution: | (1) Department of Mathematics and Statistics, Carleton University, K1S 5B6 Ottawa, Ontario, Canada |
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Abstract: | We prove that, given an arbitrary spread out probability measure μ on an almost connected locally compact second countable
groupG, there exists a homogeneous spaceG/H, called the μ-boundary, such that the space of bounded μ-harmonic functions can be identified withL
∞
(G/H). The μ-boundary is an amenable contractive homogeneous space. We also establish that the canonical projection onto the μ-boundary
of the right random walk of law μ always converges in probability and, whenG is amenable, it converges almost surely. The μ-boundary can be characterised as the largest homogeneous space among those
homogeneous spaces in which the canonical projection of the random walk converges in probability. |
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Keywords: | |
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