Spectral analysis of random walk operators on euclidian space

TL;DR

This paper analyzes the spectral properties of random walk operators on Euclidean space, providing detailed spectral descriptions near 1 and estimating convergence rates to stationarity, including Gaussian cases.

Contribution

It offers a precise spectral analysis of random walk operators on Euclidean space, extending to Gaussian densities, and estimates convergence to equilibrium.

Findings

Spectral description near 1 for the operator

Estimates of total variation distance convergence

Application to Gaussian densities

Abstract

We study the operator associated to a random walk on $\R^d$ endowed with a probability measure. We give a precise description of the spectrum of the operator near $1$ and use it to estimate the total variation distance between the iterated kernel and its stationary measure. Our study contains the case of Gaussian densities on $\R^d$.