Random walk in mixed random environment without uniform ellipticity

Ostap Hryniv, Mikhail V. Menshikov, and Andrew R. Wade

Proceedings of the Steklov Institute of Mathematics, 282, no. 1, October 2013, 106–123. DOI: 10.1134/S0081543813060102

Abstract

We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) 'fast points' with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed ('stable') random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to determine the behaviour of the random walk; we identify phase transitions in terms of the model parameters. We give conditions for recurrence and transience, and prove almost-sure bounds for the trajectories of the walk.