Markov Properties for Random Fields

Abstract

This chapter discusses the Markov properties for Gaussian generalized random fields and stochastic processes. After preliminaries involving definitions, the chapter presents relations of the Markov property to local operators, quasi-invariant measures, structure of reproducing kernel Hubert spaces, and differential equations. The chapter highlights conditional independence and its consequences, Markov properties for multi-parameter processes, Markov property on a class of sets, Markov properties for generalized random fields, quasi-invariant measures and the Markov property, Markov property for generalized Gaussian random fields, and Markov property of solutions of stochastic partial differential equations.