Characterization of continuous stationary fields as generalized Ornstein-Uhlenbeck fields via multi-parameter Langevin equation and multiple Riemann-Stieltjes integration

Abstract

In this article, we characterize continuous stationary fields via generalized Langevin dynamics. This gives natural connections between stationary fields, stationary increment fields, self-similar fields, and generalized Langevin dynamics. Our contribution extends some recently proved similar results for stochastic processes to the case of continuous random fields. As a by-product, we introduce some new results on multiple Riemann-Stieltjes integrals.