On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape
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| Format: | Preprint |
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2022
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| _version_ | 1866909395046105088 |
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| author | Castro, Matheus M. Goverse, Vincent P. H. Lamb, Jeroen S. W. Rasmussen, Martin |
| author_facet | Castro, Matheus M. Goverse, Vincent P. H. Lamb, Jeroen S. W. Rasmussen, Martin |
| contents | In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $μ$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq η\in L^1(M,μ)$. We find conditions on the transition densities of $\mathcal P$ with respect to $μ$ which ensure that $η(x) μ(\mathrm d x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $μ$-almost surely. We apply this result to the random logistic map $X_{n+1} = ω_n X_n (1-X_n)$ absorbed at $\mathbb R \setminus [0,1],$ where $ω_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_01281 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape Castro, Matheus M. Goverse, Vincent P. H. Lamb, Jeroen S. W. Rasmussen, Martin Probability Dynamical Systems 37H05, 47B65, 60J05 In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $μ$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq η\in L^1(M,μ)$. We find conditions on the transition densities of $\mathcal P$ with respect to $μ$ which ensure that $η(x) μ(\mathrm d x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $μ$-almost surely. We apply this result to the random logistic map $X_{n+1} = ω_n X_n (1-X_n)$ absorbed at $\mathbb R \setminus [0,1],$ where $ω_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$ |
| title | On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape |
| topic | Probability Dynamical Systems 37H05, 47B65, 60J05 |
| url | https://arxiv.org/abs/2209.01281 |