Passage-times for partially-homogeneous reflected random walks on the quadrant

da Costa, Conrado; Menshikov, Mikhail V.; Wade, Andrew R.

Abstract

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries. Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which the recurrence question was studied recently by Peigné and Woess (Ann. Appl. Probab. 31, 2021) using different methods, but for which no previous quantitative results on passage-times appear to be known.

Citation

da Costa, C., Menshikov, M. V., & Wade, A. R. (2025). Passage-times for partially-homogeneous reflected random walks on the quadrant. Electronic Journal of Probability, 30, 1-34. https://doi.org/10.1214/25-EJP1313

• Journal Article Type: Article
• Acceptance Date: Mar 10, 2025
• Online Publication Date: Mar 28, 2025
• Publication Date: 2025
• Deposit Date: Jul 14, 2023
• Publicly Available Date: Mar 31, 2025
• Journal: Electronic Journal of Probability
• Electronic ISSN: 1083-6489
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 30
• Article Number: 55
• Pages: 1-34
• DOI: https://doi.org/10.1214/25-EJP1313
• Public URL: https://durham-repository.worktribe.com/output/1168466