Variance-sensitive Perron bounds and Poisson equations for finite Markov chains

Variance-sensitive Perron bounds and Poisson equations for finite Markov chains Author: Zihuan XuSchool of Mathematical Science, Capital Normal UniversityORCID: 0009-0002-1510-8754License: CC-BY-4.0 This paper studies bounded additive functionals of finite irreducible Markov chains using the least singular value of the discrete generator on mean-zero L2(mu). It proves a variance-sensitive Perron pressure bound, a stationary finite-time estimate with an explicit Perron-vector prefactor, and the corresponding exponential rate for each fixed chain. Nonnormal countdown chains describe pointwise Perron and Poisson behavior at uniformly positive gap. Statewise Poisson and conditional-variance bounds yield a finite-time Bernstein inequality, with coefficient 1/32 and an explicit asymptotic variance for the centered indicator of a directed-excursion family. The final section formulates a uniform stationary scalar-power question. The text credits the generator singular-value and concentration literature and the bounded-martingale method. An AI assistant supported exploration, retrieval, English drafting, LaTeX preparation and finite rational transcription checks, as disclosed in the manuscript.