Isotropic Gauss - Markov Currents *

A natural definition of the Markov property for multi-parameter random processes (random fields) is the following. Let {Xt, tEIR N} be a multiparameter process. For any set D in N. N, let a D denote the a-field generated by {Xt, tED}. The field {Xt, tEN. u} is said to be Markov (or Markov of degree 1 [6], or sharp Markov) if, for any bounded open set D with smooth boundary, oD and ape are conditionally independent given aOD. 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.