Markov processes and their functional in duality

Abstract

Let X and X be a pair of Markov processes in duality. Formally, this means that their transition functions are adjoint operators on L ~ of some measure space. The interpretation in terms of their paths is that X is the process X somehow run backward in time. The formal duality alone is quite powerful--after establishing some elementary results, one can prove quite amazing things just by manipulating formulae--but we want to look at the processes more from the point of view of their sample paths. In fact we backslide a bit rather quickly: the main result of w 2 is a formula which relates in a simple way quite complicated properties of the two processes, and most of the calculations in the following sections are based on it. It does make more precise in what sense X is "X run backward", however, and suggests a way to relate the functionals of one to the functionals of the other. If, for instance, A is an additive functional of one process, one can construct explicitly an additive functional A of the other which is dual to A in a quite natural sense. Because the construction is explicit, it is straightforward--given the form of the construction, "straightbackward" might be a better word--to deduce the properties of A from those of A. We consider only additive and multiplicative functionals in detail, but the method extends to other cases. The first four sections concern right continuous processes in duality and hence both A and its dual .4 must be right continuous. One could just as well consider a situation where X and A were right continuous but the dual process 2 and dual functional A were left continuous. In fact, in view of some of the complications introduced in making A right continuous, this might be not only easier, but more natural in some sense. In this spirit, the last section investigates duality without imposing continuity hypotheses. We see there that if the sample paths of one process have right limits, those of its dual must necessarily have left limits, and conversely. Let E be a locally compact separable metric space and A a point isolated from E. Let g be the topological Borel field of E u A, and b g the set of bounded g-measurable functions on E w A which vanish at A. ~2 is the space of all functions from R + to E u A which are right continuous, have left limits, and admit a lifetime ~; that is, co (t) + A if t ~ (co). As usual, we designate the coordinate variable by X~(co)=co(t), (we will often use Xt instead of X t for clarity) and let (4 ~ be the natural Borel fields of the process (Xt), with ~o = V 4 ~ Now suppose that we have two Markov processes X and X with semigroups (P~) and (4) respectively, defined canonically on ~2. This means that for each x, the semigroup (Pt) generates a measure px on (f2, ~o), and that the canonical