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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 Sér. 6, 24 no. 4: Numéro Spécial : Conférence “Talking Across Fields” du 24 au 28 mars 2014 à l’Institut de Mathématiques de Toulouse (2015), p. 837-855, doi: 10.5802/afst.1467
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Résumé - Abstract

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 $\mu $ such that, for any $\epsilon >0$, $\sum |\cdot |^\epsilon \mu =\infty $. In particular, we provide a sharp lower bound for the return probability in the case when $\mu $ has a finite weak-logarithmic moment.


[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 2902834 |  Zbl 1251.60004
[2] Bendikov (A.) and Saloff-Coste (L.).— Random walks driven by low moment measures, Ann. Probab. 40, no. 6, p. 2539-2588 (2012).  MR 3050511 |  Zbl 1262.60005
[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 898871 |  Zbl 0617.26001
[4] de la Harpe (P.).— Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (2000).  MR 1786869 |  Zbl 0965.20025
[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 723729 |  Zbl 0539.60022
[6] Gromov (M.).— Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., no. 53, p. 53-73 (1981). Numdam |  MR 623534 |  Zbl 0474.20018
[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 1217561 |  Zbl 0776.60086
[8] Jacob (N.).— Pseudo differential operators and Markov processes. Vol. I, Imperial College Press, London, 2001, Fourier analysis and semigroups.  MR 1873235 |  Zbl 1076.60003
[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 1817783 |  Zbl 0985.60043
[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 3126576
[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 2978140 |  Zbl 1257.33001
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