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

Part of: Markov processes Special classes of linear operators Random dynamical systems

Published online by Cambridge University Press:  25 September 2023

MATHEUS M. CASTRO Open the ORCID record for MATHEUS M. CASTRO [Opens in a new window] ,
VINCENT P. H. GOVERSE Open the ORCID record for VINCENT P. H. GOVERSE [Opens in a new window] ,
JEROEN S. W. LAMB Open the ORCID record for JEROEN S. W. LAMB [Opens in a new window]  and
MARTIN RASMUSSEN Open the ORCID record for MARTIN RASMUSSEN [Opens in a new window]
MATHEUS M. CASTRO*
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK (e-mail: vincent.goverse21@imperial.ac.uk, jsw.lamb@imperial.ac.uk, m.rasmussen@imperial.ac.uk)
VINCENT P. H. GOVERSE
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK (e-mail: vincent.goverse21@imperial.ac.uk, jsw.lamb@imperial.ac.uk, m.rasmussen@imperial.ac.uk)
JEROEN S. W. LAMB
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK (e-mail: vincent.goverse21@imperial.ac.uk, jsw.lamb@imperial.ac.uk, m.rasmussen@imperial.ac.uk) International Research Center for Neurointelligence, The University of Tokyo, Tokyo 113-0033, Japan Centre for Applied Mathematics and Bioinformatics, Department of Mathematics and Natural Sciences, Gulf University for Science and Technology, Halwally, Kuwait
MARTIN RASMUSSEN
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK (e-mail: vincent.goverse21@imperial.ac.uk, jsw.lamb@imperial.ac.uk, m.rasmussen@imperial.ac.uk)
*
e-mail: m.de-castro20@imperial.ac.uk
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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 independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$

Keywords

Markov chains with absorptionrandom dynamical systemsquasi-stationary measurequasi-ergodic measureYaglom limit

MSC classification

Primary: 37H05: Foundations, general theory of cocycles, algebraic ergodic theory 60J05: Discrete-time Markov processes on general state spaces
Secondary: 47B65: Positive operators and order-bounded operators
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 38
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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