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

2025/11/14 by Voutilainen, Marko, Ilmonen, Pauliina, Viitasaari, Lauri
#26B99 #60G05 #60G10 #60G18 #60G60 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2511.11110

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.

Citations

• One-to-one correspondences between discrete multivariate stationary, self-similar and stationary increment fields
• On Lamperti transformation and characterisations of discrete random fields
• Modeling and estimation of multivariate discrete and continuous time stationary processes
• Gaussian self-similar random fields with distinct stationary properties of their rectangular increments
• Representation of stationary and stationary increment processes via\n Langevin equation and self-similar processes
• Example of a Gaussian self-similar field with stationary rectangular\n increments that is not a fractional Brownian sheet
• Least squares estimator for the parameter of the fractional Ornstein-Uhlenbeck sheet
• Operator Scaling Stable Random Fields

Related

• Representation of stationary and stationary increment processes via\n Langevin equation and self-similar processes
• On Lamperti transformation and characterisations of discrete random fields
• One-to-one correspondences between discrete multivariate stationary, self-similar and stationary increment fields
• G-Gaussian random fields and stochastic quantization under nonlinear expectation
• The second fluctuation-dissipation theorem for the generalized Langevin equation