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[We study a random walk (Markov chain) in an unbounded planar domain bounded by two curves of the form x2=a+x1β+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x_2 = a^+ x_1^{\beta ^+}$$ \end{document} and x2=−a−x1β−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x_2 = -a^- x_1^{\beta ^-}$$ \end{document}, with x1 ≥ 0. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle α+ or α− to the relevant inwards-pointing normal vector. Here we focus on the case where α+ and α− are equal but opposite, which includes the case of normal reflection. For 0 ≤ β+, β− < 1, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.]
Published: Nov 4, 2020
Keywords: Reflected random walk; Generalized parabolic domain; Recurrence; Transience; Passage-time moments; Normal reflection; Oblique reflection; 60J05 (Primary); 60J10; 60G50 (Secondary)
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