Invariant power law distribution of Langevin systems with colored multiplicative noise

The random multiplicative process is studied for the case of a colored mult iplicative noise with exponentially decreasing autocorrelation function. We observe the power law exponent of probability distribution in a statistica lly steady state numerically to clarify the effect of finite correlation ti me. The renormalization procedure is applied to derive the power law expone nt theoretically. The power law exponent is inversely proportional to the a utocorrelation time of the multiplicative noise.