Evolution of the system with multiplicative noise

The governed equations for the order parameter, one- and two-time correlato rs are obtained for systems with white multiplicative noise. We consider th e noise whose amplitude depends on stochastic variable as x(a) where 0 < a < 1. It turns out that the equation for autocorrelator includes an anomalou s average of the power-law function with the fractional exponent 2a. Determ ination 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 disord ered, the correlator behaves in the course of time non-monotonically, where as the autocorrelator increases monotonically. At a < 1/2 the phase portrai t of the system divides into two domains: at small initial values of the or der parameter, the system evolves to a disordered state, as above; within t he ordered domain it is attracted to the point with finite values of the au tocorrelator and order parameter. The long-time asymptotes are defined to s how that, within the disordered domain, the autocorrelator decays hyperboli cally and the order parameter behaves as a power-law function with fraction al exponent -2(1 - a). Correspondingly, within the ordered domain, the beha viour of both dependencies is exponential with an index proportional to -t ln t. <(c)> 2001 Elsevier Science B.V. All rights reserved.