Random Evolution and Geometric Probabilities

Abstract

This chapter is mainly concerned with enlarging the scope of the stochastic analysis to the description of the “random evolution” of discrete media and the application of geometric probabilities. In the preceding sections (1.5, 2.3) most of the models used in the phenomenology of discrete media were based on homogeneous Markov processes in which the transition probabilities were time independent and displayed the semi-group properties. In many applications however, the “distribution of states” is also required in the representation of the deformational or flow behaviour of the media, in particular, if interaction effects between structural elements are to be included. Evidently, this necessitates the use of a wider class of Markov processes usually involving “jump processes”, partially observed jump processes and also “semi-Markov processes”.