MARKOV-FIELDS OVER COUNTABLE PARTIALLY ORDERED SETS - EXTREMA AND SPLITTING

Various notions of Markov property relative to a partial ordering have been proposed by both physicists and mathematicians. For the most par t, the analysis of these notions has been focussed on the study of som e important, but special examples. In this work, we develop general te chniques for studying Markov fields on partially ordered sets. In part icular, we introduce and examine certain classes of ''random time chan ges'', random transformations of the index set which preserve the Mark ov property of the field. These transformations yield new classes of M arkov fields, starting from more basic or simple ones. Random elements of the index set involved in the ''random time changes'' are similar to spitting times in the theory of stochastic processes. We construct such random elements as solutions to stochastic extremal problems rela ted to the field. The general results are illustrated in a variety of random field models which have physical interpretation. In particular, we consider models of crack formation and models of fibre composite m aterials. In these models, the extremal problems involved in the const ructions of splitting random elements have a clear physical meaning. T hey describe energetically optimal processes in random environment.