The exit characteristics of Markov processes with applications to continuous martingales in Rⁿ

Abstract

Introduction. The main results of this paper were announced in [4]. A weaker form of the results in ??1 and 2 appeared in the author's thesis(1) [5], the research for which was supported by the National Science Foundation. Paul Levy showed in [11] that Brownian motion on the line can be characterized in terms of two martingale functions. Recently Arbib [1] extended this result to include all diffusions on the line with natural boundaries at infinity. His method was to characterize a diffusion by its exit characteristics, but his proof was only of use in the specific cases he was considering. In ?1 we obtain, by probabilistic methods, a similar characterization of minimal (no return from the boundary) right continuous strong Markov processes on a general state space. In ?2 we use this to generalize Levy's theorem to Rn, and in ?4 indicate some extension of Arbib's results to Rn. In ?3 we make use of our general Levy-type theorem to obtain a characterization of all continuous martingales in Rn, extending a result of Dambis [3] on the line. It was recently brought to my attention that Kunita and Watanabe [10] have used quite different methods to obtain somewhat weaker results which were also, in essence, reported by Dubins and Schwarz [7], who used still another method of proof.