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We study a one-dimensional random walk whose expected drift depends both ontime and the position of a particle. We establish a non-trivial phasetransition for the recurrence vs. transience of the walk, and show someinteresting applications to Friedman's urn, as well as showing the connectionwith Lamperti's walk with asymptotically zero drift.

Urn-related random walk with drift $\rho x^{\alpha} / t^{\beta}$