Gauss-Markov random fields (CMrf) with continuous indices

Gauss-Markov random fields (GMrfs) play an important role in the modeling of physical phenomena. The paper addresses the second-order characterization and the sample path description of GMrf´s when the indexing parameters take values in bounded subsets of ℜ^d, d⩾1. Using results of Pitt (1994), we give conditions for the covariance of a GMrf to be the Green´s function of a partial differential operator and, conversely, for the Green´s function of an operator to be the covariance of a GMrf. We then develop a minimum mean square error representation for the field in terms of a partial differential equation driven by correlated noise. The paper establishes for GMrf´s on ℜ^d second-order characterizations that parallel the corresponding results for GMrf´s on finite lattices