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 $0\leq \eta \in L^1(M,\mu )$ . We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which en...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2024-07, Vol.44 (7), p.1818-1855
Main Authors: CASTRO, MATHEUS M., GOVERSE, VINCENT P. H., LAMB, JEROEN S. W., RASMUSSEN, MARTIN
Format: Article
Language:English
Subjects:
Dynamical systems
Eigenvalues
Eigenvectors
Ergodic processes
Hypotheses
Independent variables
Markov analysis
Markov chains
Original Article
Random variables
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Summary: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 independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a 4.$
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2023.69