An extended Markov property

Introduction. Let x (t) ; /SiOJ be a time homogeneous Markov process. A positive random variable T defined on the same underlying probability space as Jx (/) j is called a Markov time for the process if the collection of functions {x (t+T) ;/SiOj also forms a Markov process with the same transition function as the original process and if, when x (7") is fixed, the new process is independent of the original one considered only up to time T. Thus a time homogeneous Markov process could be defined as a process for which T is a Markov time whenever T is identically a constant. This paper is chiefly concerned with finding hypotheses on {x (Z) and on T which will insure that T be a Markov time. Certain related matters including the behavior of sample functions are also discussed. Whatever loose statements are made in this introduction will be made precise at the appropriate places in the body of the paper. In probabilistic approaches to potential theory (Doob l, Hunt l; 2) and in other researches which employ the familiar "first passage time relationship" the random variables T which arise are the "first passage times" or more general "stopping times" which will be defined below; so the question of whether these are Markov times is of especial interest. Hunt l has shown that if x (t) is a process in w-space with stationary independent increments then any stopping time is a Markov time for the process provided x (t) has right continuous sample functions (such processes can generally be normalized to have this right continuity property according to a theorem of Doob 2). Hunt states his result in the following form: {x (t+T) -x (T) ; /SiOj is a Markov process with the same transition function as the original process and is completely independent of the original process considered only up to time T. He also points out that this result is valid for processes taking values in more general spaces. We shall see that for a somewhat more general class of processes any stopping time is a Markov time, although the theorem for these processes can not be expressed in the same form as Hunt's result. Among these processes are the time homogeneous processes of Ito in which a drift and a spread, varying in space but not in time, are imposed upon the Brownian motion.