A Fuzzy Decomposition Method to Establish Functional Inequalities

TL;DR

This paper introduces a fuzzy decomposition approach to establish functional inequalities for Markov chains, allowing for more flexible partitions of the state space and easier estimation of inequalities like Poincaré and log-Sobolev.

Contribution

It generalizes previous methods by using fuzzy partitions, enabling the analysis of chains where exact partitions are not feasible, thus broadening applicability.

Findings

Fuzzy decomposition effectively estimates functional inequalities.

The method applies to cases where exact partitions are not possible.

Simplifies analysis of Markov chains with complex state spaces.

Abstract

This paper presents a method to establish functional inequalities via fuzzy decomposition on the state space, which generalizes earlier results dealing with exact partitions of the state space. Given a reversible Markov chain on a finite state space, we define its projection chain and restriction chains from classes of a fuzzy partition on the state space. The Poincar\'e, log-Sobolev and modified log-Sobolev inequalities associated with the original chain can be estimated from those of its projection chain and restriction chains which tend to have simpler structures and hence easier to work with. An application of this generalization is presented in which the fuzzy decomposition method applies but its exact partitioning counterpart does not.