Iv. Evstigneev et Pe. Greenwood, MARKOV-FIELDS OVER COUNTABLE PARTIALLY ORDERED SETS - EXTREMA AND SPLITTING, Memoirs of the American Mathematical Society, 112(537), 1994, pp. 1
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.