Long range random walks and associated geometries on groups of polynomial growth
Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 1249-1304.

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study the large time behavior of its probability of return at time n in terms of the key parameters describing the driving measure and the structure of the underlying group. We obtain assorted estimates including near-diagonal two-sided estimates and the Hölder continuity of the solutions of the associated discrete parabolic difference equation. In each case, these estimates involve the construction of a geometry adapted to the walk.

Dans le contexte des groupes finiment engendrés à croissance polynomiale du volume, nous considérons une large classe de marches aléatoires à sauts de longue portée distribués suivant des lois puissances dans la direction de plusieurs sous-groupes. Pour de telles marches, nous déterminons la probabilité de retour au temps n en fonction de la distribution des sauts et de la structure algébrique du groupe. Nous obtenons des estimations autour de la diagonale ainsi que la continuité Hölderienne des solutions de l’équation de la chaleur discrète associée. Dans chaque cas, ces estimations utilisent la géométrie associée à la marche.

Published online:
DOI: 10.5802/aif.3515
Classification: 60G50, 20F65, 60B15
Keywords: long range random walk, group, return probability, Pseudo-Poincaré inequality, Hölder continuity
Mot clés : Marches aléatoires à sauts non bornés, groupe, probabilité de retour, inégalité pseudo-Poincaré, continuité Hölderienne
Chen, Zhen-Qing 1; Kumagai, Takashi 2; Saloff-Coste, Laurent 3; Wang, Jian 4; Zheng, Tianyi 5

1 University of Washington Department of Mathematics Seattle, WA 98195 (USA)
2 Kyoto University Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502 (Japan)
3 Cornell University Department of Mathematics Ithaca, NY 14853 (USA)
4 Fujian Normal University School of Mathematics and Statistics & Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA) & Center for Applied Mathematics of Fujian Province (FJNU) Fuzhou 350007 (P.R. China)
5 UC San Diego Department of Mathematics San Diego, CA 92093-0112 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Long range random walks and associated geometries on groups of polynomial growth},
     journal = {Annales de l'Institut Fourier},
     pages = {1249--1304},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Chen, Zhen-Qing; Kumagai, Takashi; Saloff-Coste, Laurent; Wang, Jian; Zheng, Tianyi. Long range random walks and associated geometries on groups of polynomial growth. Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 1249-1304. doi : 10.5802/aif.3515. https://aif.centre-mersenne.org/articles/10.5802/aif.3515/

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