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 journalArticlepeer-review

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 languageEnglish
Pages (from-to)1249-1304
Number of pages56
JournalAnnales de l'Institut Fourier
Volume72
Issue number3
DOIs
Publication statusPublished - 2022
Externally publishedYes

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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