Characterization of continuous stationary fields as generalized Ornstein-Uhlenbeck fields via multi-parameter Langevin equation and multiple Riemann-Stieltjes integration
Marko Voutilainen, Pauliina Ilmonen, Lauri Viitasaari
TL;DR
The paper extends the Langevin framework to continuous multi-parameter random fields, providing a dynamic representation of stationary fields via a generalized multi-parameter Langevin equation and linking them to self-similar and stationary-increment fields. It develops a robust multi-parameter Riemann-Stieltjes calculus, including integration over nonstandard domains, and uses Lamperti-type transformations to establish bijections between stationary fields, self-similar fields, and fields with stationary rectangular increments. The main contributions are a Langevin-based characterization of continuous stationary fields with driving processes in a structured class, a rigorous RS-integral foundation for multi-dimensional settings, and the identification of fractional Ornstein–Uhlenbeck fields as special cases. Together, these results provide a rigorous dynamic representation and analytical toolkit for analyzing spatial-temporal random fields across dimensions, with potential applications in modeling complex phenomena exhibiting temporal and spatial dependencies.
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.