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 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 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.
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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
@article{AIF_2022__72_3_1249_0, author = {Chen, Zhen-Qing and Kumagai, Takashi and Saloff-Coste, Laurent and Wang, Jian and Zheng, Tianyi}, 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}, volume = {72}, number = {3}, year = {2022}, doi = {10.5802/aif.3515}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3515/} }
TY - JOUR AU - Chen, Zhen-Qing AU - Kumagai, Takashi AU - Saloff-Coste, Laurent AU - Wang, Jian AU - Zheng, Tianyi TI - Long range random walks and associated geometries on groups of polynomial growth JO - Annales de l'Institut Fourier PY - 2022 SP - 1249 EP - 1304 VL - 72 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3515/ DO - 10.5802/aif.3515 LA - en ID - AIF_2022__72_3_1249_0 ER -
%0 Journal Article %A Chen, Zhen-Qing %A Kumagai, Takashi %A Saloff-Coste, Laurent %A Wang, Jian %A Zheng, Tianyi %T Long range random walks and associated geometries on groups of polynomial growth %J Annales de l'Institut Fourier %D 2022 %P 1249-1304 %V 72 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3515/ %R 10.5802/aif.3515 %G en %F AIF_2022__72_3_1249_0
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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