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

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 $0\le \eta \in L^1(M,\mu )$. 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 $1\le a<4$ and $b>4.$