Random walk in a birth-and-death dynamical environment
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909444922671104 |
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| author | Fontes, Luiz Renato Gomes, Pablo Almeida Pinheiro, Maicon Aparecido |
| author_facet | Fontes, Luiz Renato Gomes, Pablo Almeida Pinheiro, Maicon Aparecido |
| contents | We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on ${\mathbb Z}^d$ , and its jump rate at $({\mathbf x},t)$ is given by a fixed function $φ$ of the state of a birth-and-death (BD) process at $\\mathbf x$ on time $t$; BD processes at different sites are independent and identically distributed, and $φ$ is assumed non increasing and vanishing at infinity. We derive a LLN and a CLT for the particle position when the environment is 'strongly ergodic'. In the absence of a viable uniform lower bound for the jump rate, we resort instead to stochastic domination, as well as to a subadditive argument to control the time spent by the particle to give $n$ jumps; and we also impose conditions on the initial (product) environmental initial distribution. We also present results on the asymptotics of the environment seen by the particle (under different conditions on $φ$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_02154 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Random walk in a birth-and-death dynamical environment Fontes, Luiz Renato Gomes, Pablo Almeida Pinheiro, Maicon Aparecido Probability 60K37, 60F05 We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on ${\mathbb Z}^d$ , and its jump rate at $({\mathbf x},t)$ is given by a fixed function $φ$ of the state of a birth-and-death (BD) process at $\\mathbf x$ on time $t$; BD processes at different sites are independent and identically distributed, and $φ$ is assumed non increasing and vanishing at infinity. We derive a LLN and a CLT for the particle position when the environment is 'strongly ergodic'. In the absence of a viable uniform lower bound for the jump rate, we resort instead to stochastic domination, as well as to a subadditive argument to control the time spent by the particle to give $n$ jumps; and we also impose conditions on the initial (product) environmental initial distribution. We also present results on the asymptotics of the environment seen by the particle (under different conditions on $φ$). |
| title | Random walk in a birth-and-death dynamical environment |
| topic | Probability 60K37, 60F05 |
| url | https://arxiv.org/abs/2211.02154 |