Abstract
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 Bauman, Bass and Burdzy, and Fabes et al., based on comparison arguments and potential-theoretical tools; the existence of a positive $\tilde{L}$-harmonic function globally defined in the half-space; and some Gaussian inequalities obtained by a treatment inspired by Varopoulos.
Citation
Sami Mustapha. "Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments." Bernoulli 13 (1) 131 - 147, February 2007. https://doi.org/10.3150/07-BEJ5135
Information