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

TL;DR

This paper addresses the problem of deriving the Kipnis–Varadhan functional central limit theorem for additive functionals of reversible ergodic Markov processes by specializing Bhattacharya's general fclt to the time-reversible case. The authors leverage the resolvent identity $f=\lambda R_\lambda f-\hat{A}R_\lambda f$ and decompose $I_n(f,t)$ into a vanishing part and a main part, controlling the vanishing part through a sequence $\lambda_n\to 0$ with $\lambda_n=o(1/n)$. For fixed small $\lambda_\ell$, the main term converges to a Brownian motion with dispersion $\sigma_{\lambda_\ell}^2 = 2\langle -\hat{A}R_{\lambda_\ell}f, R_{\lambda_\ell}f\rangle_\pi$, which increases to the target variance $\sigma^2(f)=2\langle(-\hat{A})^{-1/2}f,(-\hat{A})^{-1/2}f\rangle_\pi$ as $\lambda_\ell\to 0$, yielding the Kipnis–Varadhan FCLT for $f \in \mathcal{D}_{(-\hat{A})^{-1/2}}$. The result clarifies the interchange of limits and provides a constructive route via semigroup resolvents and Yosida theory, with implications for solute dispersion and interacting particle systems.

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.

Theorems & Definitions (8)

• Theorem 1: Bhattacharya RBfclt
• Remark 1
• Remark 2
• Theorem 2: Yosida's Potential Operator yosidakomatsu
• Proposition 1
• Corollary 1
• Theorem 3: Kipnis-Varadhan $\mathcal{D}_{(-\hat{A})^{-\frac{1}{2}}}$ Condition
• proof