Strong Markov properties for Markov random fields

Markov properties and strong Markov properties for random fields are define d and discussed. Special attention is given to those defined by I. V. Evsti gneev. The strong Markov nature of Markov random fields with respect to ran dom domains such as [0, L], where l is a multidimensional extension of a st opping time is explored. A special case of this extension is shown to gener alize a result of Merzbach and Nualart for point processes. As an additiona l example, Evstigneev's Markov and strong Markov properties are considered for independent increment jump processes.