In this work, we consider nearest neighbour q-random walks on Z^d for d = 1,2,3, with transition probabilities q-varying by the number of steps, 0 < q < 1, and we study under which conditions these q-random walks are transient or recurrent. Also, we define the relative continuous time q-random walks on the integers and on the two dimensional integer lattices. Moreover, we present a q-Brownian motion as a continuous analogue of the q-random walk on the integers and we study the maxima and first hitting time of this q-Brownian motion. Furthermore, we present simulations, using R statistical computing environment, of the considered stochastic processes on Z^d for d = 1,2 and of the q-Brownian motion, indicating first hitting times to the origin.

在这项工作中，我们考虑最近邻q -random行走ž ^d为d = 1,2,3，用转移概率q -varying通过的步数，0 < q <1，我们研究在哪些条件下这些q -随机行走是短暂的或反复发生的。同样，我们定义了相对连续时间q-随机游走在整数和二维整数晶格上。此外，我们提出了一个q-布朗运动，作为q-随机游动在整数上的连续模拟，我们研究了该q的最大值和第一次撞击时间-布朗运动。此外，我们提出了使用R统计计算环境对d = 1,2的Z ^d上考虑的随机过程和q-布朗运动进行的模拟，这表明到原点的首次命中时间。