Theory Of Markov Processes | E. B. Dynkin

The present book aims at investigating the logical founda- tions of the theory of Markov random processes.

The theory of Markov processes has developed rapidly in recent years, The properties o£ the trajectories of such processes and their infinitesimal operators have been studied, and intimate connexions have been discovered

between the behaviour of the trajectories and the properties of the differential equations corresponding to the process. These connexions are useful for studying differential equa- tions as well as Markov processes. The material thus accumu- lated has made necessary a critical survey of the fundamen- tals of the theory. In particular, the usual statement o£ the Markov principle of "absence of after-effects" has been found to be inadequate and various authors have proposed different forms for a strengthened principle whereby a pro- cess is "strictly Markov." It has become obvious that the most natural subject for s~~dy is presented by Markov pro- cesses cut of£ at a random instant. All these and other ideas were originally introduced by different authors in different forms, according to the specific purposes of their specialized works - in which stationary Markov processes are considered almost exclusively,

This book cannot be used by the student to make his first acquaintance with the theory of Markov processes. Although we have not assumed formally any previous acquaintance with the theory of probability, in fact a reading of the book can only prove of value to someone already acquainted with an elementary exposition of the theory of Markov processes, such as is contained, for instance, in Feller's "Introduc- tion to probability theory and its applications," Vol. 1 (Vvedenie v teoriyu veroyatnostei i ee prilozheniya), or Gnedenko's "Course of probability theory" (Kurs teorii veroyatnostei).