2023/12/25 by Dong, Congzao, Iksanov, Alexander, Pilipenko, Andrey
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2312.15806
Let d be a positive integer and A a set in ℤd, 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 ℤd, 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≥ 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 Lévy 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 ℤd for d≥ 3. More interestingly, in the situation where the standard random walk in ℤ2 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.