Multiplicative random walks

Simple examples of multiplicative random walks are considered in which the random variable X is multiplied by a given scaling factor at each step of the process. Several cases are analyzed, either pure or disordered, showing how the disorder can affect the variation in time of various expectation values. It is seen that unstable (inflating) exponentially diverging cases are only slightly “renormalized” by disorder, even strong. On the contrary, for the deflating regimes, increasing disorder turns the asymptotic regime from exponential into algebraic decay in time.