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Power-law distributions are ubiquitous in nature. Random multiplicativeprocesses are a basic model for the generation of power-law distributions. Itis known that, for discrete-time systems, the power-law exponent decreases asthe autocorrelation time of the multiplier increases. However, forcontinuous-time ystems, it has not yet been elucidated as to how the temporalcorrelation affects the power-law behavior. Herein, we have analyticallyinvestigated a multiplicative Langevin equation with colored noise. We showthat the power-law exponent depends on the details of the multiplicative noise,in contrast to the case of discrete-time systems.

Power-law exponent in multiplicative Langevin equation with temporally correlated noise