On multidimensional locally perturbed standard random walks

Let d be a positive integer, and let A be a set in Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}<^>{d}$$\end{document} that contains finitely many points with integer coordinates. We consider a standard random walk X perturbed on the set A. This means that X is a Markov chain whose transition probabilities from the points outside A coincide with those of a standard random walk on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}<^>{d}$$\end{document}, 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 L & eacute;vy process, or some other "skew" process. The distinction between the one-dimensional and multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}<^>{d}$$\end{document} for d >= 3. More interestingly, in the situation where the standard random walk in Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}<^>{2}$$\end{document} 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, we have 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.