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Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments. (English) Zbl 1111.62070

Summary: Generalizing to higher dimensions the classical gambler’s ruin estimates, we give pointwise estimates for the transition kernel corresponding to a spatially inhomogeneous random walk on the half-space. Our results hold under some strong but natural assumptions of symmetry, boundedness of the increments, and ellipticity. Among the most important steps in our proof are: discrete variants of the boundary Harnack estimate, as proven by P. Bauman [Ark. Mat. 22, 153–173 (1984; Zbl 0557.35033)], R. F. Bass and K. Burdzy [J. Lond. Math. Soc., II. Ser. 50, No. 1, 157–169 (1994; Zbl 0806.35025)], and E. B. Fabes et al. [Trans. Am. Math. Soc. 351, No. 12, 4947–4961 (1999; Zbl 0976.35031)], based on comparison arguments and potential-theoretical tools; the existence of a positive \(\widetilde L\)-harmonic function globally defined in the half-space; and some Gaussian inequalities obtained by a treatment inspired by N. Th. Varopoulos [see Can. J. Math. 55, No. 2, 401–431 (2003; Zbl 1042.58013)].

MSC:

62M09 Non-Markovian processes: estimation
62M05 Markov processes: estimation; hidden Markov models
60J45 Probabilistic potential theory
31C20 Discrete potential theory
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