Negatively Curved Manifolds, Elliptic Operators, and the Martin Boundary

Ancona, Alano

doi:10.2307/1971409

Cited by 191 publications

(248 citation statements)

References 15 publications

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“…It is straightforward to see that R(µ) is smooth at the two 'front faces' of the product, ff 1 ×(M 2 ) 2 0 and (M 1 ) 2 0 × ff 2 . This is because the exponents in the expansions of R 1 (µ 1 )R 2 (µ − µ 1 ) at these faces do not depend on µ 1 . It remains to analyze the behaviour at the remaining 'side faces'.…”

Section: Asymptotics Inside the Continuous Spectrummentioning

confidence: 99%

Resolvents and Martin boundaries of product spaces

Mazzeo¹,

Vasy²

2002

GAFA, Geom. funct. anal.

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Section: Asymptotics Inside the Continuous Spectrummentioning

confidence: 99%

Resolvents and Martin boundaries of product spaces

Mazzeo¹,

Vasy²

2002

GAFA, Geom. funct. anal.

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“…The complete description of the Poisson-Furstenberg boundary has been known for the following finitely generated groups (under certain conditions on the decay of the probability measure defining the random walk): discrete subgroups in semi-simple Lie group (Furstenberg [27] for a particular case of an infinitely supported measure, "Furstenberg approximation", Ledrappier [50] for the case of discrete subgroups of SL.d; R/, Kaimanovich [40] for a general class of measures), free groups (Dynkin, Malyutov [18] for simple random walk on standard generators, Derriennic [14] for measures with finite support), more generally for hyperbolic groups (Ancona [1] for measures with finite support, Kaimanovich [40] for measures of finite entropy and with finite logarithmic moments; see also [5]), groups with infinitely many ends (Woess [58] for finitely supported measures, [40] for more general class of measures), the mapping class group (Kaimanovich,Masur [42]), braid groups (Farb,Masur [23]), for wreath products of free groups with finite groups (Karlsson, Woess [47]), Coxeter groups (follows from Karlsson, Margulis [46], see Theorem 6.1 in [45] for an explanation). Sometimes it is easier to identify the boundary for certain nonsymmetric random walks, rather than for symmetric ones.…”

Section: Theorem 1 Let Be An Adapted Measure Onmentioning

confidence: 99%

Poisson–Furstenberg boundary of random walks on wreath products and free metabelian groups

Erschler¹

2010

Comment. Math. Helv.

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Abstract. We study the Poisson-Furstenberg boundary of random walks on C D A o B, where A D Z d and B is a finitely generated group having at least 2 elements. We show that for d 5, for any measure on C such that its third moment is finite and the support of the measure generates C as a group, the Poisson boundary can be identified with the limit "lamplighter" configurations on A. This provides a partial answer to a question of Kaimanovich and Vershik [44]. Also, for free metabelian groups S d;2 on d generators, d 5, we answer a question of Vershik [56] and give a complete description of the Poisson-Furstenberg boundary for any non-degenerate random walk on S d;2 having finite third moment. Finally, we give various examples of slowly decaying measures on wreath products with non-standard boundaries.

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“…It is easy to see that a SRW is a 1/N -NRW with c(x, x − ) = c(x, y) = 1 for (x, y) ∈ E h . Using Theorem 1.1 and a well-known result of Ancona [1,2,44] on uniformly irreducible random walks on hyperbolic graphs, we prove the following theorem (Theorem 5.1) which extends [25,Theorem 4.7] for the Sierpiński graph with the simple random walk. Theorem 1.2.…”

Section: For a Homogenous Ifs {Smentioning

confidence: 84%

“…The following important result is due to Ancona [1,2], and the specific version we use is taken from [44, Theorem 27.1].…”

Section: Preliminariesmentioning

confidence: 99%

Random walks and induced Dirichlet forms on self-similar sets

Kong

Lau

Wong

2017

Advances in Mathematics

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Let K be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea [25], it has been proved that on the symbolic space X of K a natural augmented tree structure E exists; it is hyperbolic, and the hyperbolic boundary ∂ H X with the Gromov metric is Hölder equivalent to K. In this paper we consider certain reversible random walks with return ratio 0 < λ < 1 on (X, E). We show that the Martin boundary M can be identified with ∂ H X and K. With this setup and a device of Silverstein [41], we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate Θ(ξ, η) ≍ |ξ − η| −(α+β) , where α is the Hausdorff dimension of K and β depends on λ. For suitable β, the kernel defines a regular non-local Dirichlet form on K. This extends the results of Kigami [27] concerning random walks on certain trees with Cantor-type sets as boundaries (see also [5]).

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Negatively Curved Manifolds, Elliptic Operators, and the Martin Boundary

Cited by 191 publications

References 15 publications

Resolvents and Martin boundaries of product spaces

Resolvents and Martin boundaries of product spaces

Poisson–Furstenberg boundary of random walks on wreath products and free metabelian groups

Random walks and induced Dirichlet forms on self-similar sets

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