Recurrence and transience of random walks with drift $\rho x^{\alpha }/t^{\beta }$

Abstract

Menshikov and Volkov [Electron. J. Probab. 13 (2008)] studied recurrence and transience of a class of Markovian random walks on ${\mathbb{R}}_{+}$ whose conditional drift depends on both time and position and is of order $\rho x^{\alpha }{t}^{-\beta }$ with $\rho >0$. The case on the critical line $2\beta -\alpha =1$, with $\alpha \in (-1,1)\setminus \{0\}$, remained open. We prove recurrence in this remaining case. Furthermore, we establish recurrence and transience criteria that complete the classification for $-1<\alpha <1$ and $\beta \ge 0$, without assuming the Markov property and under weaker assumptions on the increments than those imposed by Menshikov and Volkov.

21 pages