Stochastic Kolmogorov systems driven by wideband noises.

In many problems arising in statistical physics, statistical mechanics, and many related fields, one needs to deal with such nonlinear stochastic differential equations as Ginzburg–Landau equations and Lotka–Volterra equations, etc. Such equations all belong to the class of stochastic Kolmogorov systems. Because of their importance and wide range of applications, these systems have received much attention in recent years. Devoted to stochastic Kolmogorov systems, in contrast to the usual setup of using a Brownian motion as a driving force, in this paper, the underlying system is assumed to be subject to wideband type of noise perturbations. The main thought is that Brownian motion is an idealization used in a wide range of applications, whereas the wideband noise processes is much easier to be realized in the actual applications. Although it is a good approximation to a diffusion process, the process under wideband noise becomes non-Markovian. Using weak convergence methods, we show that the limits are the desired Kolmogorov systems driven by Brownian motions. • In contrast to the existing work, this paper treats Kolmogorov systems under wideband noise perturbations. • This paper shows under suitable conditions, the systems under wideband noises converge weakly to systems driven by a white noise using a martingale problem formulation. • The systems under wideband noises are generally non-Markovian, which is suited for the actual systems. The limit, however, becomes Markovian. [ABSTRACT FROM AUTHOR]