On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape

TL;DR

This paper investigates the quasi-ergodic behavior of absorbing Markov chains with unbounded transition densities, including applications to random logistic maps with escape, establishing conditions for convergence to quasi-stationary measures.

Contribution

It provides new conditions on transition densities that ensure quasi-ergodicity and convergence of Yaglom limits for a class of absorbing Markov chains, including random logistic maps.

Findings

Conditions for quasi-ergodic measures are established.

Yaglom limit convergence is proven under these conditions.

Application to random logistic maps with escape at boundaries.

Abstract

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq \eta\in L^1(M,\mu)$. We find conditions on the transition densities of $\mathcal P$ with respect to $\mu$ which ensure that $\eta(x) \mu(\mathrm d x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $\mu$-almost surely. We apply this result to the random logistic map $X_{n+1} = \omega_n X_n (1-X_n)$ absorbed at $\mathbb R \setminus [0,1],$ where $\omega_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$