Chapter 8 - Gaussian delta-correlated random field (ordinary differential equations)

Abstract

This chapter presents a detailed consideration of the approximation of the Gaussian random delta-correlated (in time) field in the context of stochastic equations and discusses the physical meaning of this widely used approximation. Applicability of the approximation of the delta-correlated random field f(x,t) (i.e., applicability of the Fokker-Planck equation) is restricted by the smallness of the temporal correlation radius τ0 of random field f (x, t) with respect to all temporal scales of the problem under consideration. The effect of the finite-valued temporal correlation radius of random field f (x, t) can be considered within the framework of the diffusion approximation. The diffusion approximation appears more obvious and physical than the formal mathematical derivation of the approximation of the delta-correlated random field. This approximation also holds for sufficiently weak parameter fluctuations of the stochastic dynamic system and allows describing new physical effects caused by the finite-valued temporal correlation radius of random parameters, rather than only obtaining the applicability range of the delta-correlated approximation. The diffusion approximation assumes that the effect of random actions is insignificant during temporal scales about τ0, i.e., the system behaves during these times as the free system.