Analytical and numerical studies of multiplicative noise

TLDR

Stochastic differential equations with multiplicative white and colored noise model nonequilibrium systems experiencing non‑self‑originating fluctuations. The study aims to develop and analyze numerical and analytical methods for simulating such equations and to examine their stationary solutions. The authors present a numerical algorithm, derive approximate Fokker‑Planck equations by expanding in the noise correlation time τ, analyze stationary solutions, and apply these techniques to the Stratonovich model. Numerical results confirm the analytical predictions, showing that at high noise intensity the stationary distribution develops a dominant peak as τ grows and that the steady‑state correlation time increases with both D and τ, contradicting earlier Stratonovich analyses.

Abstract

We consider stochastic differential equations for a variable $q$ with multiplicative white and nonwhite ("colored") noise appropriate for the description of nonequilibrium systems which experience fluctuations which are not "self-originating." We discuss a numerical algorithm for the simulation of these equations, as well as an alternative analytical treatment. In particular, we derive approximate Fokker-Planck equations for the probability density of the process by an analysis of an expansion in powers of the correlation time $\ensuremath{\tau}$ of the noise. We also discuss the stationary solution of these equations. We have applied our numerical and analytical methods to the "Stratonovich model" often used in the literature to study nonequilibrium systems. The numerical analysis corroborates the analytical predictions for the time-independent properties. We show that for large noise intensity $D$ the stationary distribution develops a peak for increasing $\ensuremath{\tau}$ that becomes dominant in the large-$\ensuremath{\tau}$ limit. The correlation time of the process in the steady state has been analyzed numerically. We find a "slowing down" in the sense that the correlation time increases as a function of both $D$ and $\ensuremath{\tau}$.. .. This result shows the incorrectness of an earlier analysis of Stratonovich.