On multidimensional locally perturbed standard random walks
| Autor: | Dong, Congzao, Iksanov, Alexander, Pilipenko, Andrey |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $d$ be a positive integer and $A$ a set in $\mathbb{Z}^d$, which contains finitely many points with integer coordinates. We consider $X$ a standard random walk perturbed on the set $A$, that is, a Markov chain whose transition probabilities from the points outside $A$ coincide with those of a standard random walk on $\mathbb{Z}^d$, whereas the transition probabilities from the points inside $A$ are different. We investigate the impact of the perturbation on a scaling limit of $X$. It turns out that if $d\geq 2$, then in a typical situation the scaling limit of $X$ coincides with that of the underlying standard random walk. This is unlike the case $d=1$ in which the scaling limit of $X$ is usually a skew Brownian motion, a skew stable L\'{e}vy process or some other `skew' process. The distinction between the one-dimensional and the multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in $\mathbb{Z}^d$ for $d\geq 3$. More interestingly, in the situation where the standard random walk in $\mathbb{Z}^2$ is recurrent, the preservation of its Donsker scaling limit is secured by the fact that the number of visits of $X$ to the set $A$ is much smaller than in the one-dimensional case. As a consequence, the influence of the perturbation vanishes upon the scaling. On the other edge of the spectrum is the situation in which the standard random walk admits a Donsker's scaling limit, whereas its locally perturbed version does not because of huge jumps from the set $A$ which occur early enough. Comment: 18 pages, submitted for publication |
| Databáze: | arXiv |
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