Z. Wahrscheinlichkeitstheorie verw. Gebiete 48, 97-114 (1979) Zeitschrift far Wahrscheinlichkeitstheorie und verwandteGebiete 9 by Springer-Verlag 1979 Dual Markov Functionals: Applications of a Useful Auxiliary Process Joanna B. Mitro Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA O. Introduction In a companion paper [11] we constructed a two-sided time homogeneous Markov process Z, starting from a given pair of Markov processes (Q, ~, ~,~, X, 0 t, px) and (~, ~,~ J~, )~, 0,, fix) on the state space E which are in duality relative to a a-finite measure ~. Under our regularity assumptions the auxiliary process Z may be taken to be the coordinate mapping Z,(~)=o(t) on the space f2 of paths from 1R into E(A, A) which are right continuous with left limits and admit A as birth point and A as "cemetery", with corresponding a- fields ~~ On the measurable space (~,~~ c~)) we define the a-finite measure Q so that on {ZteE}, s~Zt+ s is equivalent under (2 to X s with initial distribution ~, and s-+Zt_ , is equivalent under Q to )~s under /~. In this paper we use the auxiliary process Z to give a new solution to the problem of construction of dual additive and multiplicative functionals and apply our techniques to define a duality theory for a type of functional recently introduced into the study of Markov processes: the comultiplicative functional E2]. The existence of dual additive and multiplicative functionals is well estab- lished. Given a multiplicative functional M (right continuous, decreasing, with 0=~ M ~ 1) there is an associated resolvent (V ~) corresponding to the subordinate semigroup (Kt)"generated" by M, defined by K,f(x)=EX(f(Xt) 9 M,) for f~bg +. Under the hypothesis of duality a resolvent (V~) dual to (V ~) can be constructed which corresponds to a dual semigroup (R,), subordinate to (/~) and generated from (/~) by a multiplicative functional 2~([4, 9]). M and 2~r are said to be dual multiplicative functionals, and satisfy (0.1) ~ K,f(x) g(x) ~ (dx) = ~f(x) g/~,(x) ~ (dx) for f gebo ~+. The probabilistic interpretation of (0.1) is that the two "killed" subprocesses (X, M) and (Js ~r) are in duality relative to ~. The situation for additive functionals is slightly more complicated. (See Revuz [12], Getoor [3], and Sharpe [13] for a description of the analytic 0044-3719/79/0048/0097/$03.60