Random walk in mixed random environment without uniform ellipticity

Ostap Hryniv ; Mikhail V. Menshikov ; Andrew R. Wade

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 114-131.

Résumé

We study a random walk in random environment on $\mathbb Z_+$. 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.

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