Long range random walks and associated geometries on groups of polynomial growth

TL;DR

This paper investigates long-range random walks on groups with polynomial growth, analyzing their return probabilities and associated geometries, and providing estimates and continuity results for the related difference equations.

Contribution

It introduces a framework for studying long-range random walks on polynomial growth groups, including new estimates and geometric constructions tailored to these walks.

Findings

Derived near-diagonal two-sided estimates for return probabilities

Established Hölder continuity of solutions to associated difference equations

Constructed geometries adapted to the long-range random walks

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\"older 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.