Generalized Markov fields and Dirichlet forms

Abstract

We prove that Gaussian measure-indexed random fields, of which the covariance functional is given by the dual form of a transient Dirichlet form, have the global Markov property (where ‘global’ here means ‘w.r.t. arbitrary, not necessarily open sets’), if and only if the Dirichlet form has the local property. Applications to Nelson's free Euclidean field of quantum theory and to Rozanov's generalized random functions are given.