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

TL;DR

This paper characterizes continuous stationary fields using generalized Langevin dynamics, establishing connections with various types of fields and extending prior process results to random fields, while also introducing new findings on multiple Riemann-Stieltjes integrals.

Contribution

It extends results from stochastic processes to continuous random fields and introduces new insights into multiple Riemann-Stieltjes integrals.

Findings

Connections between stationary fields and Langevin dynamics established

Extension of process results to random fields achieved

New results on multiple Riemann-Stieltjes integrals introduced

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.