Hitting Time Distributions for General Stochastic Processes

Abstract

It is the purpose of this paper to suggest, by a simple example, that the methods which have been so successful in the study of temporally homogeneous Markov processes can be applied equally successfully to general stochastic processes. Let i be a probability measure in the space Q, W) of all measures on the measurable space (Q, SW) of all functions wt) mapping R+ = (0, oo) into a measurable space (S, ?) where _ is the a-field generated by the events X,(wO) = w(t) C U C ? and where S is a separable compact space with Borel sets ?, and suppose that the continuous functions in Q have i-outer measure one. Let Tt, t C R+ be the semigroup of linear operators on (Q, ) defined by Tje(X1l C U1, *..., IXtn CUn) = [(Xt+tl C U1,***, Xt+tn C U.) and let Eu, U C ? be the resolution of the identity Eu M(A) = [(Xo C U, A). Let $ be the weak * closure over the continuous functions on the product topology of (Q, SW) of the linear subspace of (Q, S) which is generated by measures of the form Eun Ttn ... EU1 Tt1 i and let $+ be the set of all probability measures in O. Suppose now that T is the first exit time of X from the interior U of S and let g be a continuous function on the boundary U' of U. Let (* be the set of all linear functionals 0* on ( which are continuous in the weak * topology of O. Let Tt*o*o = O*Tto 0* C (* 0 D and G*0*0= limh oh-l[Th*0*0 00* e @ 0 e +