Spectral theoretic characterisation of Markov chain convergence

TL;DR

This paper introduces a spectral theoretic approach to analyze Markov chain convergence by linking their statistics to differential operators, providing explicit convergence rates and applicability to various chaotic maps.

Contribution

It presents a novel spectral method that characterizes Markov chain convergence and invariant measures through associated differential operators, offering explicit convergence rates.

Findings

Successfully characterizes Markov chain statistics via differential operators.

Provides explicit convergence rates to invariant measures.

Demonstrates applicability to logistic, tent, and Chebyshev maps.

Abstract

In this work, we characterise the statistics of Markov chains by constructing an associated sequence of periodic differential operators. Studying the density of states of these operators reveals the absolutely continuous invariant measure of the Markov chain. This approach also leads to a direct proof of convergence to the invariant measure, along with explicit convergence rates. We show how our method can be applied to a class of related Markov chains including the logistic map, the tent map and Chebyshev maps of arbitrary order.