Comultiplicative functionals and the birthing of a Markov process

Abstract

It has been known for many years that a multiplicative functional, re, may be used to kill a Markov process. A special case of this construction is when m is given by a terminal time T as m t -1E0 , r)(t). More recently Meyer, Smythe, and Walsh I-8] introduced the notion of a coterminal time and showed how a coterminal time may be used to "b i r th" a Markov process. In view of this it is natural to ask if there is a p r o c e s s naturally called a comultiplicative func t iona l which has the same relationship to coterminal times as a multiplicative functional has to terminal times. After some preliminaries in Section 2 we introduce the notion of a comultiplicative functional in Section 3, Definition (3.1). In Sections 3 and 5 we show that there is a complete duality between comultiplicative functionals and an appropriate class of multiplicative functionals. In the case of coterminal times this reduces to the duality with terminal times given in [8]. In Section 4 we show how a comultiplicative functional may be used to birth a Markov process in a manner that is dual to that by which a multiplicative functional is used to kill a process. Sections 4 and 5 are independent of each other and may be read in either order. What we develop here might be called an "algebraic" theory since we assume the exceptional sets in our definitions are empty. In light of the recent work of Walsh 1-9] and Meyer [7] this causes no problems in dealing with multiplicative functionals. However, the a-algebras that we use are motivated by the results of Meyer [7]. Obviously there are dual perfection properties for comultiplicative functionals, but we do not discuss this here. I hope to return to it in a future publication. The duality developed in Sections 3 and 5 does not involve a Markov process. It may be viewed as a chapter in the duality between shift (Is birth better?) operators (0t) and killing operators (kt) developed by various authors in recent years. See, especially, Azema [1]. The Markov process, itself, enters only in Section 4. In 1-8] Meyer, Smythe, and Walsh also showed that a cooptional time may be used to kill a process. There is an analogous result here for cooptional functionals that is, a functional n satisfying only condition (3.1)(i) of Definition (3.1). How-