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On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 7 / July 2024
- Published online by Cambridge University Press:
- 25 September 2023, pp. 1818-1855
- Print publication:
- July 2024
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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 
$b>4.$
