Isotropic Gauss-Markov currents

Abstract

A natural definition of the Markov property for multi-parameter random processes (random fields) is the following. Let {Xt,t∈ℝN} be a multiparameter process. For any set D in ℝN let σD denote the σ-field generated by {Xt, t∈D}. The field {Xt,t∈D} is said to be Markov (or Markov of degree 1 [6], or sharp Markov) if, for any bounded open set D with smooth boundary, σD and σDc are conditionally independent given σδD. It has been known for some time that to find interesting examples of Markov processes under this definition; it is necessary to consider generalized random functions. In this paper we show that a natural framework for the Markov property of multiparameter processes is a class of generalized random differential forms (i.e., random currents). Our principal objective is to relate the Markovian nature of an isotropic gaussian current to its spectral properties.