We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in terms of certain continued fractions such that this stochastic factorization is always possible. By inverting the order of the factors (also known as a Darboux transformation) we get new families of random walks on the integers. We identify the spectral matrices associated with these Darboux transformations (in both cases) which are basically conjugations by a matrix polynomial of degree one of a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk with constant transition probabilities with or without an attractive or repulsive force.

我们考虑整数上随机游动的跃迁概率矩阵的UL和LU随机因式分解，它是一个无限大的三对角随机Jacobi矩阵。我们根据某些连续分数给出两个分解的自由参数的条件，使得这种随机分解总是可能的。通过反转因子的顺序（也称为Darboux变换），我们获得了新的整数随机游动族。我们确定了与这些Darboux转换（在两种情况下）相关的光谱矩阵，它们基本上是原始光谱矩阵的Geronimus变换的一阶矩阵多项式的共轭。最后，我们将结果应用于具有或不具有吸引力或排斥力的具有恒定过渡概率的随机游动。