A set-indexed Ornstein-Uhlenbeck process

TL;DR

This paper introduces a set-indexed Ornstein-Uhlenbeck process, extending the classical model to a broader context with new properties and integral representations, enhancing the understanding of stochastic processes indexed by sets.

Contribution

It provides a comprehensive set-indexed extension of the Ornstein-Uhlenbeck process, including a complete characterization, Markov properties, and integral representation in the multiparameter case.

Findings

Characterization by $L^2$-continuity, stationarity, and Markov properties.

Definition of a general set-indexed Ornstein-Uhlenbeck process with any initial measure.

Existence of a natural integral representation in the multiparameter case.

Abstract

The purpose of this article is a set-indexed extension of the well-known Ornstein-Uhlenbeck process. The first part is devoted to a stationary definition of the random field and ends up with the proof of a complete characterization by its $L^2$-continuity, stationarity and set-indexed Markov properties. This specific Markov transition system allows to define a general \emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial probability measure. Finally, in the multiparameter case, the SIOU process is proved to admit a natural integral representation.