Random walks in the group of Euclidean isometries and self-similar measures

TL;DR

This paper investigates random isometries in Euclidean space, establishing a local limit theorem with exponential convergence and demonstrating absolute continuity of certain self-similar measures with smooth densities, extending Bernoulli convolution results.

Contribution

It proves a local limit theorem for random isometries with spectral gap assumptions and shows that specific self-similar measures are absolutely continuous with smooth densities, generalizing Bernoulli convolutions.

Findings

Proved a local limit theorem with exponential speed for random isometries.

Established absolute continuity and smooth densities for certain self-similar measures.

Extended Bernoulli convolution results to higher dimensions.

Abstract

We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov operator associated to the rotation component of the isometries has spectral gap. We also prove that certain self-similar measures are absolutely continuous with smooth densities. These families of self-similar measures give higher dimensional analogues of Bernoulli convolutions on which absolute continuity can be established for contraction ratios in an open set.