Spartan Random Fields and Langevin Equations

Abstract

In this chapter we look at Spartan random fields from a different perspective. Our first goal is to show that solutions of linear stochastic partial differential (Langevin) equations are random fields with rational spectral densities [694]. In addition, the respective covariance function is the Green’s function of a suitable (i.e., derivable from Langevin equation) partial differential equation. Finally, the joint dependence of random fields that satisfy Langevin equations driven by a Gaussian white noise process can be expressed in terms of an exponential Gibbs-Boltzmann pdf; the latter has a quadratic energy function that involves local (i.e., based on low-order field derivatives) terms.