Poisson Boundaries of Random Walks on Discrete Solvable Groups

Vadim A. Kaimanovich
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Vadim A. Kaimanovich: Leningrad Shipbuilding Institute

A chapter in Probability Measures on Groups X, 1991, pp 205-238 from Springer

Abstract: Abstract Let G be a topological group, and μ — a probability measure on G. A function f on G is called harmonic if it satisfies the mean value property % MathType!MTEF!2!1!+- % feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI % cacaWGNbGaaiykaiabg2da9maapeaabaGaamOzaiaacIcacaWGNbGa % amiEaiaacMcacaWGKbGaeqiVd0MaaiikaiaadIhacaGGPaaaleqabe % qdcqGHRiI8aaaa!4546! !!! $$ f(g) = \int {f(gx)d\mu (x)} $$ for all g ∈ G. It is well known that under natural assumptions on the measure μ there exists a measure G-space Γ with a quasi-invariant measure v such that the Poisson formula % MathType!MTEF!2!1!+- % feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI % cacaWGNbGaaiykaiabg2da9iabgYda8iqadAgagaqcaiaacYcacaWG % NbGaamODaiabg6da+aaa!3FC7! !!! $$ f(g) = $$ states an isometric isomorphism between the Banach space H ∞(G, μ) of bounded harmonic functions with sup-norm and the space X∞(Γ, μ). The space (Γ, v) is called the Poisson boundary of the pair (G, μ). Thus triviality of the Poisson boundary is equivalent to absence of non-constant bounded harmonic functions for the pair (G, μ) (the Liouville property).

Keywords: Random Walk; Semidirect Product; Extended Chain; Solvable Group; Wreath Product (search for similar items in EconPapers)
Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4899-2364-6_16

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DOI: 10.1007/978-1-4899-2364-6_16

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