Concentration of Additive Functionals for Markov Processes and Applications to Interacting Particle Systems

TL;DR

This paper develops new concentration bounds for additive functionals of Markov processes using coupling methods, applicable to diffusions, particle systems, and random walks without requiring reversibility.

Contribution

It introduces a coupling-based approach to derive concentration bounds for Markov processes, avoiding spectral theory and reversibility assumptions.

Findings

Derived exponential moment bounds for additive functionals

Established relations between semigroup contractivity and coupling time

Applicable to diffusions, particle systems, and random walks

Abstract

We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions, interacting particle systems and random walks. The method is based on coupling estimates and not spectral theory, hence reversibility is not needed. We bound the exponential moments(or the moments of finite order) in terms of a so-called coupled function difference, which in turn is estimated using the generalized coupling time. Along the way we prove a general relation between the contractivity of the semigroup and bounds on the generalized coupling time.