Long range random walks and associated geometries on groups of polynomial growth
Annales de l'Institut Fourier, Online first, 56 p.

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.

Received:
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Accepted:
Online First:
DOI: 10.5802/aif.3515
Classification: 60G50,  20F65,  60B15
Keywords: long range random walk, group, return probability, Pseudo-Poincaré inequality, Hölder continuity
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)
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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, Online first, 56 p.

[1] Bass, Richard F.; Chen, Zhen-Qing Regularity of harmonic functions for a class of singular stable-like processes, Math. Z., Volume 266 (2010) no. 3, pp. 489-503 | DOI | MR | Zbl

[2] Bass, Richard F.; Kumagai, Takashi Symmetric Markov chains on d with unbounded range, Trans. Am. Math. Soc., Volume 360 (2008) no. 4, pp. 2041-2075 | DOI | MR

[3] Bass, Richard F.; Levin, David A. Transition probabilities for symmetric jump processes, Trans. Am. Math. Soc., Volume 354 (2002) no. 7, pp. 2933-2953 | DOI | MR | Zbl

[4] Bingham, Nicholas H.; Goldie, Charles M.; Teugels, Jozef L. Regular variation, Encyclopedia of Mathematics and Its Applications, 27, Cambridge University Press, 1987, xx+491 pages | MR

[5] Chaker, Jamil; Kassmann, Moritz Nonlocal operators with singular anisotropic kernels, Commun. Partial Differ. Equations, Volume 45 (2020) no. 1, pp. 1-31

[6] Chen, Zhen-Qing; Kumagai, Takashi Heat kernel estimates for stable-like processes on d-sets, Stochastic Processes Appl., Volume 108 (2003) no. 1, pp. 27-62 | DOI | MR

[7] Chen, Zhen-Qing; Lou, Shuwen Brownian motion on spaces with varying dimension, Ann. Probab., Volume 47 (2019) no. 1, pp. 213-269

[8] Coulhon, Thierry Ultracontractivity and Nash type inequalities, J. Funct. Anal., Volume 141 (1996) no. 2, pp. 510-539 | DOI | MR

[9] Coulhon, Thierry; Saloff-Coste, Laurent Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoam., Volume 9 (1993) no. 2, pp. 293-314 | MR

[10] Delmotte, Thierry Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoam., Volume 15 (1999) no. 1, pp. 181-232

[11] Doob, Joseph L. Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer, 2001, xxvi+846 pages (Reprint of the 1984 edition) | DOI | MR

[12] Dynkin, Evgeniĭ B.; Malyutov, Mikhail B. Random walk on groups with a finite number of generators, Dokl. Akad. Nauk SSSR, Volume 137 (1961), pp. 1042-1045 | MR

[13] Erschler, Anna; Zheng, Tianyi Growth of periodic Grigorchuck groups, Invent. Math., Volume 219 (2020) no. 3, pp. 1069-1155

[14] Grigorchuk, Rostislav Milnor’s problem on the growth of groups and its consequences, Frontiers in complex dynamics (Princeton Mathematical Series), Volume 51, Princeton University Press, 2014, pp. 705-773

[15] Hebisch, Waldemar; Saloff-Coste, Laurent Gaussian estimates for Markov chains and random walks on groups, Ann. Probab., Volume 21 (1993) no. 2, pp. 673-709

[16] Kaimanovich, Vadim A.; Vershik, Anatoliĭ M. Random walks on discrete groups: boundary and entropy, Ann. Probab., Volume 11 (1983) no. 3, pp. 457-490

[17] Kesten, Harry Symmetric random walks on groups, Trans. Am. Math. Soc., Volume 92 (1959) no. 2, pp. 336-354

[18] Margulis, Grigoriĭ A Positive harmonic functions on nilpotent groups, Sov. Math., Dokl., Volume 7 (1966), pp. 241-244 | MR

[19] Peres, Yuval; Zheng, Tianyi On groups, slow heat kernel decay yields Liouville property and sharp entropy bounds, Int. Math. Res. Not. (2020) no. 3, pp. 722-750

[20] Pittet, Christophe; Saloff-Coste, Laurent On the stability of the behavior of random walks on groups, J. Geom. Anal., Volume 10 (2000) no. 4, pp. 713-737 | MR

[21] Saloff-Coste, Laurent Probability on groups: random walks and invariant diffusions, Notices Am. Math. Soc., Volume 48 (2001) no. 9, pp. 968-977 | MR

[22] Saloff-Coste, Laurent; Zheng, Tianyi Random walks on nilpotent groups driven by measures supported on powers of generators, Groups Geom. Dyn., Volume 9 (2015) no. 4, pp. 1047-1129 | DOI | MR

[23] Saloff-Coste, Laurent; Zheng, Tianyi Random walks and isoperimetric profiles under moment conditions, Ann. Probab., Volume 44 (2016) no. 6, pp. 4133-4183 | DOI | MR

[24] Saloff-Coste, Laurent; Zheng, Tianyi On some random walks o driven by spread-out measures, Groups, graphs and random walks (London Mathematical Society Lecture Note Series), Volume 436, Cambridge University Press, 2017, pp. 444-474

[25] Varopoulos, Nicholas Th. Théorie du potentiel sur les groupes nilpotents, C. R. Math. Acad. Sci. Paris, Volume 301 (1985) no. 5, pp. 143-144 | MR

[26] Varopoulos, Nicholas Th.; Saloff-Coste, Laurent; Coulhon, Thierry Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, 1992, xii+156 pages | MR

[27] Woess, Wolfgang Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138, Cambridge University Press, 2000, xii+334 pages | MR

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