Geodesic
Random walk in mixed random environment without uniform ellipticity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 114-131

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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. 
@article{TM_2013_282_a9,
     author = {Ostap Hryniv and Mikhail V. Menshikov and Andrew R. Wade},
     title = {Random walk in mixed random environment without uniform ellipticity},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {114--131},
     publisher = {mathdoc},
     volume = {282},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_282_a9/}
}
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Ostap Hryniv; Mikhail V. Menshikov; Andrew R. Wade. Random walk in mixed random environment without uniform ellipticity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 114-131. http://geodesic.mathdoc.fr/item/TM_2013_282_a9/