Chapter 7 - Gaussian random field delta-correlated in time (ordinary differential equations)

Abstract

This chapter provides an alternative and more detailed consideration of the approximation of the Gaussian random delta-correlated field in the context of stochastic equations and discusses the physical meaning of this widely used approximation. A vector function that satisfies the dynamic equation is presented. Statistical characteristics of field are completely described by the correlation tensor. It is shown that the one-time probability density of solution at instant is governed by functional dependence of solution on field for all times in the interval. The Fokker–Planck equation is a partial differential equation and its further analysis essentially depends on boundary conditions. It is found that to estimate the applicability range of the Fokker–Planck equation, we must include into consideration the finite-valued correlation radius of field with respect to time. It is emphasized that the approximation of the delta-correlated random field does not reduce to the formal replacement of random field with the random field with correlation function.