The Brownian Motion on Aff(R) and Quasi-Local Theorems

TL;DR

This paper studies random walk approximations of Brownian motion on the affine group Aff(R), showing that integrating return probabilities near the origin restores polynomial decay, leading to a quasi-local theorem.

Contribution

It introduces a quasi-local theorem demonstrating how integrating return probabilities recovers polynomial decay in the discrete approximation of Brownian motion on Aff(R).

Findings

Return probabilities decay fractionally exponentially for discrete innovations.

Integrating return probabilities near the origin restores polynomial decay.

The quasi-local theorem links discrete walk behavior to continuous Brownian motion properties.

Abstract

This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probability of the walk have fractional exponential decay in large time, as opposed to the polynomial one of the continuous object. We prove that integrating those return probabilities on a suitable neighborhood of the origin, the expected polynomial decay is restored. This is what we call a Quasi-local theorem.