Gauss-Markov random fields (CMrf) with continuous indices

Gauss–Markov random fields (GMrf’s) 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 <; d 1. Using results of Pitt, 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 < secondorder characterizations that parallel the corresponding results for GMrf’s on finite lattices.