Evolution of the System with Singular Multiplicative Noise

Abstract

The governed equations for the order parameter, one-time and two-time correlators are obtained on the basis of the Langevin equation with the white multiplicative noise which amplitude $x^{a}$ is determined by an exponent $0<a<1$ ($x$ being a stochastic variable). It turns out that equation for autocorrelator includes an anomalous average of the power-law function with the fractional exponent $2a$. Determination of this average for the stochastic system with a self-similar phase space is performed. It is shown that at $a>1/2$, when the system is disordered, the correlator behaves non-monotonically in the course of time, whereas the autocorrelator is increased monotonically. At $a<1/2$ the phase portrait of the system evolution divides into two domains: at small initial values of the order parameter, the system evolves to a disordered state, as above; within the ordered domain it is attracted to the point having the finite values of the autocorrelator and order parameter. The long-time asymptotes are defined to show that, within the disordered domain, the autocorrelator decays hyperbolically and the order parameter behaves as the power-law function with fractional exponent $-2(1-a)$. Correspondingly, within the ordered domain, the behavior of both dependencies is exponential with an index proportional to $-t\ln t$.