An Invariance Principle for a Random Walk Among Moving Traps via Thermodynamic Formalism
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arXiv
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| Format: | Preprint |
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2024
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| author | Athreya, Siva Drewitz, Alexander Sun, Rongfeng |
| author_facet | Athreya, Siva Drewitz, Alexander Sun, Rongfeng |
| contents | We consider a random walk among a Poisson cloud of moving traps on ${\mathbb Z}^d$, where the walk is killed at a rate proportional to the number of traps occupying the same position. In dimension $d=1$, we have previously shown that under the annealed law of the random walk conditioned on survival up to time $t$, the walk is sub-diffusive. Here we show that in $d\geq 6$ and under diffusive scaling, this annealed law satisfies an invariance principle with a positive diffusion constant if the killing rate is small. Our proof is based on the theory of thermodynamic formalism, where we extend some classic results for Markov shifts with a finite alphabet and a potential of summable variation to the case of an uncountable non-compact alphabet. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Invariance Principle for a Random Walk Among Moving Traps via Thermodynamic Formalism Athreya, Siva Drewitz, Alexander Sun, Rongfeng Probability Dynamical Systems 60K37, 60K35, 82C22 We consider a random walk among a Poisson cloud of moving traps on ${\mathbb Z}^d$, where the walk is killed at a rate proportional to the number of traps occupying the same position. In dimension $d=1$, we have previously shown that under the annealed law of the random walk conditioned on survival up to time $t$, the walk is sub-diffusive. Here we show that in $d\geq 6$ and under diffusive scaling, this annealed law satisfies an invariance principle with a positive diffusion constant if the killing rate is small. Our proof is based on the theory of thermodynamic formalism, where we extend some classic results for Markov shifts with a finite alphabet and a potential of summable variation to the case of an uncountable non-compact alphabet. |
| title | An Invariance Principle for a Random Walk Among Moving Traps via Thermodynamic Formalism |
| topic | Probability Dynamical Systems 60K37, 60K35, 82C22 |
| url | https://arxiv.org/abs/2401.10179 |