LONG RANGE RANDOM WALKS AND ASSOCIATED GEOMETRIES ON GROUPS OF POLYNOMIAL GROWTH

Zhen Qing Chen; Takashi Kumagai; Laurent Saloff-Coste; Jian Wang; Tianyi Zheng

doi:10.5802/aif.3515

LONG RANGE RANDOM WALKS AND ASSOCIATED GEOMETRIES ON GROUPS OF POLYNOMIAL GROWTH

Zhen Qing Chen, Takashi Kumagai, Laurent Saloff-Coste, Jian Wang, Tianyi Zheng

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	1249-1304
Number of pages	56
Journal	Annales de l'Institut Fourier
Volume	72
Issue number	3
DOIs	https://doi.org/10.5802/aif.3515
Publication status	Published - 2022
Externally published	Yes

Keywords

Hölder continuity
Pseudo-Poincaré inequality
group
long range random walk
return probability

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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10.5802/aif.3515

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title = "LONG RANGE RANDOM WALKS AND ASSOCIATED GEOMETRIES ON GROUPS OF POLYNOMIAL GROWTH",

abstract = "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 H{\"o}lder 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.",

keywords = "H{\"o}lder continuity, Pseudo-Poincar{\'e} inequality, group, long range random walk, return probability",

author = "Chen, {Zhen Qing} and Takashi Kumagai and Laurent Saloff-Coste and Jian Wang and Tianyi Zheng",

note = "Funding Information: Keywords: long range random walk, group, return probability, Pseudo-Poincar{\'e} inequality, H{\"o}lder continuity. 2020 Mathematics Subject Classification: 60G50, 20F65, 60B15. (*) The research of ZC is supported by Simons Foundation Grant 520542, a Victor Klee Faculty Fellowship at UW, and NNSFC grant 11731009, TK by the Grant-in-Aid for Scientific Research (A) 17H01093, Japan, LSC by NSF grants DMS-1404435 and DMS-1707589, and JW by NNSFC grant 11831014 and 12071076, and the Education and Research Support Program for Fujian Provincial Agencies. Publisher Copyright: {\textcopyright} 2022 Association des Annales de l'Institut Fourier. All rights reserved.",

year = "2022",

doi = "10.5802/aif.3515",

language = "English",

volume = "72",

pages = "1249--1304",

journal = "Annales de l'Institut Fourier",

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AU - Chen, Zhen Qing

AU - Kumagai, Takashi

AU - Saloff-Coste, Laurent

AU - Wang, Jian

AU - Zheng, Tianyi

N1 - Funding Information: Keywords: long range random walk, group, return probability, Pseudo-Poincaré inequality, Hölder continuity. 2020 Mathematics Subject Classification: 60G50, 20F65, 60B15. (*) The research of ZC is supported by Simons Foundation Grant 520542, a Victor Klee Faculty Fellowship at UW, and NNSFC grant 11731009, TK by the Grant-in-Aid for Scientific Research (A) 17H01093, Japan, LSC by NSF grants DMS-1404435 and DMS-1707589, and JW by NNSFC grant 11831014 and 12071076, and the Education and Research Support Program for Fujian Provincial Agencies. Publisher Copyright: © 2022 Association des Annales de l'Institut Fourier. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Hölder continuity

KW - Pseudo-Poincaré inequality

KW - group

KW - long range random walk

KW - return probability

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LONG RANGE RANDOM WALKS AND ASSOCIATED GEOMETRIES ON GROUPS OF POLYNOMIAL GROWTH

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Keywords

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