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.