Characteristic functional approach to multiplicative fractal noise with application to environmental fluctuations in nonlinear chemical systems far from equilibrium:: A temporal analog of Anderson localization

We compute the characteristic functional of a nonlinear transformation of a set of random variables. The equation is applied to the study of fractal stochastic processes with multiplicative noise. For illustration we investigate two types of chemical systems subjected to environmental fluctuations. We analyze the interaction between the intrinsic chemical fluctuations and environmental noise and evaluate the stochastic properties of the system in the presence of multiplicative noise. For an autocatalytic reaction subjected to external noise and with Gaussian and generally non-Markovian multiplicative external noise the averaged joint probability density of concentrations displays long tails of the negative power law type, which are independent of the detailed properties of the noise. We also study the influence of multiplicative noise on a nonlinear chemical system with a stable limit cycle. In the presence of multiplicative noise the limit cycle collapses and the average trajectory in the composition space becomes a spiral. This effect is a temporal analog of Anderson localization. We evaluate the damping factors and localization times for different types of noise ranging from noise with infinite memory (static disorder) and noise with long memory (dynamic disorder with fractal noise) to Markovian noise (dynamic disorder with short memory).