Central limit theorems for additive functionals of ergodic Markov diffusions processes

TL;DR

This paper revisits functional central limit theorems for additive functionals of ergodic Markov diffusion processes, providing new conditions and insights applicable to degenerate and slow diffusions within a PDE framework.

Contribution

It introduces tractable conditions on the generator for CLTs in ergodic diffusions, including degenerate cases, and connects these results to PDE analysis and recent ergodic theory developments.

Findings

Established CLT conditions for a broad class of diffusions

Extended results to degenerate and slow diffusions

Discussed open problems and applications in PDE context

Abstract

We revisit functional central limit theorems for additive functionals of ergodic Markov diffusion processes. Translated in the language of partial differential equations of evolution, they appear as diffusion limits in the asymptotic analysis of Fokker-Planck type equations. We focus on the square integrable framework, and we provide tractable conditions on the infinitesimal generator, including degenerate or anomalously slow diffusions. We take advantage on recent developments in the study of the trend to the equilibrium of ergodic diffusions. We discuss examples and formulate open problems.