On Markov Processes with Right-Deterministic Germ Fields

0. Introduction. Let (Q, , Yt, X(t), Ot, Px) be a Hunt process on a locally compact space with countable base (E, e), where e denotes the Borel sets. The notation is that of [1] page 45. For each t, let 7? denote the sigma-field generated by X(s), s 0). The influence of chance in such a process in general occurs continuously in time, but the work of K. L. Chung, P. A. Meyer, and others has shown that there is a considerable difference between the operation of chance "from the past" at time T and "to the future." To be more precise, let J/(T, T + s) be the sigma-field generated by X(T + s), 0 O JW3?[T, T + e] be the "right germ field" at time T, containing but not necessarily equaling the sigma-field a(X(T)) generated by X(T). The operation of chance to the future at T is only made possible by the non-equality of V+(T) and a(X(T)), while that from the past is still more problematical due to the lack of a really satisfactory concept of left germ field (except at constant times). However, to study the distinction between these two local effects a natural idea is to exclude one and then determine what remains of the other. The idea of the present paper is simply to exhibit the role of chance from the past by assuming that it does not exist to the future.