Chapter 9 - Gaussian delta-correlated random field (causal integral equations)

Abstract

This chapter reviews Gaussian delta-correlated random field. The problems described in terms of integral equations that generally cannot be reduced to a system of differential equations can also satisfy the causality condition. In this case, the parent stochastic equation is the linear integral equation for Green's function. It is observed that solution is a functional of field that appears equivalent to the functional equation that contains the variational derivative in functional space. It is considered that random field is the Gaussian random field whose average value is zero. In this case, assignment of the correlation function describes all statistical characteristics of the field. If one tends the temporal correlation radius of random field to zero, then the equation is simplified and assumes the form of the closed integral equation. This result is equivalent to the introduction of the effective correlation function of random field and the use of equation that just corresponds to the delta-correlated approximation for random field in time. The splitting correlators between the Gaussian delta-correlated process and functional of the process are analyzed. It is found that for the one-dimensional causal equation, the ladder approximation appears to be of the exact equality in the case of the delta-correlated process.