Markov Properties for Certain Random Fields

Abstract

Lévy's Markov and sharp Markov properties for random fields arestudied, first in a general setting, and then in the context of twoparameter processes. It is shown that if Lévy's Markov property holds relative to finite unions of sets in some neighborhood base, then it holds for all bounded open sets. Two-parameter Gaussian processes which satisfy the usual Markov property along certain oneparameter curves are shown to satisfy Lévy's Markov property; they are in fact transforms of the Brownian sheet. Finally, a new proof is given that the Poisson sheet satisfies Lévy's sharp Markov property relative to all bounded relatively convex open sets.