On Central Limit Theorems for Additive Functionals of Reversible Ergodic Markov Processes

TL;DR

This paper demonstrates that a general theorem by Bhattacharya implies the Kipnis-Varadhan CLT for ergodic Markov processes in the reversible case, using semigroup theory tools.

Contribution

It extends Bhattacharya's theorem to establish the Kipnis-Varadhan CLT specifically for reversible ergodic Markov processes.

Findings

Theorem linking Bhattacharya's result to Kipnis-Varadhan CLT in the reversible case.

Incorporation of semigroup theory, including the resolvent identity, into the proof.

Clarification of the conditions under which the CLT holds for additive functionals.

Abstract

In this note, the time reversible case of a general theorem of Bhattacharya is shown to imply the Kipnis-Varadhan functional central limit theorem for ergodic Markov processes. To this end, a few results from semigroup theory, including the resolvent identity, are incoporated in Bhattacharya's range condition for the inifinitesimal generator.