Central limit theorem under a general nonreversible condition

Prove a central limit theorem for additive functionals of general nonreversible Markov processes under a condition such as finiteness of the symmetrized \(H_{-1}\) norm, \(\|f\|_{-1,S}<\infty\).

Background

The Kipnis–Varadhan theorem is established in the reversible setting, where finiteness of the appropriate $H_{-1}$ norm yields a central limit theorem for additive functionals. The notes explain that in nonreversible systems the variance may be finite even when the symmetrized $H_{-1}$ norm is infinite.

The unresolved question is whether a general nonreversible analogue can be obtained under a natural sufficient condition, specifically finiteness of the $H_{-1}$ norm associated with the symmetric part of the generator.