Random walks under slowly varying moment conditions on groups of polynomial volume growth

Laurent Saloff-Coste; Tianyi Zheng

doi:10.5802/afst.1467

Laurent Saloff-Coste ¹ ; Tianyi Zheng ²

¹ Department of Mathematics, Cornell University
² Department of Mathematics, Stanford University

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 4, pp. 837-855.

Résumé
Abstract

Soit $G$ un groupe finiment engendré, à croissance polynomiale du volume et muni de la distance des mots associée à un ensemble donné de générateurs. Le but de ce travail est de développer des techniques qui permettent l’étude de marches aléatoires associées à des mesures de probabilité symetriques, $μ$ , telles que, pout tout $ϵ > 0$ , ${\sum | \cdot |}^{ϵ} μ = \infty$ . En particulier, nous donnons une borne inférieure optimale pour la probabilité de retour dans le cas où $μ$ a un moment logarithmique de type faible fini.

Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $| \cdot |$ . The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures $μ$ such that, for any $ϵ > 0$ , ${\sum | \cdot |}^{ϵ} μ = \infty$ . In particular, we provide a sharp lower bound for the return probability in the case when $μ$ has a finite weak-logarithmic moment.

DOI : 10.5802/afst.1467

Affiliations des auteurs :

Laurent Saloff-Coste ¹ ; Tianyi Zheng ²

¹ Department of Mathematics, Cornell University
² Department of Mathematics, Stanford University

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     title = {Random walks under slowly varying moment conditions on groups of polynomial volume growth},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {837--855},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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Laurent Saloff-Coste; Tianyi Zheng. Random walks under slowly varying moment conditions on groups of polynomial volume growth. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 4, pp. 837-855. doi : 10.5802/afst.1467. https://afst.centre-mersenne.org/articles/10.5802/afst.1467/

Bibliographie
Cité par

[1] Bendikov (A.) and Saloff-Coste (L.).— Random walks on groups and discrete subordination, Math. Nachr. 285, no. 5-6, p. 580-605 (2012). | MR | Zbl

[2] Bendikov (A.) and Saloff-Coste (L.).— Random walks driven by low moment measures, Ann. Probab. 40, no. 6, p. 2539-2588 (2012). | MR | Zbl

[3] Bingham (N. H.), Goldie (C. M.), and Teugels (J. L.).— Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge (1987). | MR | Zbl

[4] de la Harpe (P.).— Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (2000). | MR | Zbl

[5] Griffin (P. S.), Jain (N. C.), and Pruitt (W. E.).— Approximate local limit theorems for laws outside domains of attraction, Ann. Probab. 12, no. 1, p. 45-63 (1984). | MR | Zbl

[6] Gromov (M.).— Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., no. 53, p. 53-73 (1981). | Numdam | MR | Zbl

[7] Hebisch (W.) and Saloff-Coste (L.).— Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21, no. 2, p. 673-709 (1993). | MR | Zbl

[8] Jacob (N.).— Pseudo differential operators and Markov processes. Vol. I, Imperial College Press, London, 2001, Fourier analysis and semigroups. | MR | Zbl

[9] Pittet (Ch.) and Saloff-Coste (L.).— On the stability of the behavior of random walks on groups, J. Geom. Anal. 10, no. 4, p. 713-737 (2000). | MR | Zbl

[10] Saloff-Coste (L.) and Zheng (T.).— On some random walks driven by spread-out measures, Available on Arxiv arXiv:1309.6296 [math.PR], submitted (2012).

[11] Saloff-Coste (L.) and Zheng (T.).— Random walks and isoperimetric profiles under moment conditions, Available on Arxiv arXiv:1501.05929 [math.PR], submitted (2014).

[12] Saloff-Coste (L.) and Zheng (T.).— Random walks on nilpotent groups driven by measures supported on powers of generators, To appear in Groups, Geometry, and Dynamics (2013). | MR

[13] Schilling (R. L.), Song (R.), and Vondraček (Z.).— Bernstein functions, second ed., de Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, (2012), Theory and applications. | MR | Zbl

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