Construction of markov processes from hitting distributions

Abstract

As demonstrated in the fundamental memoir of Hunt [8], in the study of a general strong Markov process with nice path behavior hitting distributions play a very important role. For standard processes on a locally compact second countable Hausdorff space with the same family of hitting distributions, it is proved by Blumenthal, Getoor and McKean [2] (see also [1]) that they can be obtained, up to equivalence, from a single process by means of random time change. This suggests that a large part of the theory of Markov processes is intrinsically associated with hitting distributions rather than with transition probabilities. Thus the problem of constructing a process with given hitting distributions is a most natural one. This paper has resulted from an effort to extend the theorems of Knight and Orey [9] and Dawson [5], which deal with this problem. (The latter treats diffusion processes only.) In [11] we announced a theorem that is more general than the above theorems. All three assume given two ingredients for the construction. One is a family of measures on the state space, assumed to be locally compact second countable Hausdorff, that behave like the hitting distributions of a Markov process for a large class sets and are smooth. Smoothness means in our case that the measures transform continuous functions vanishing at infinity into such functions, which is weaker than that assumed by Knight and Orey and by Dawson in different ways. The other is a function g on the state space, continuous and vanishing at infinity, with g(x) meant to be the expected lifetime of the process starting at each state x. Knight and Orey's condition on g is rather unnatural, while Dawson's and ours involve the fine topology defined from the given hitting distributions. The present result is obtained by first relaxing the condition that g is required to satisfy in our earlier result, and then constructing such a function from the given hitting distributions, which now satisfy a natural transience condition in addition to the original smoothness condition. (The form of the function g is suggested in [9].) This furnishes a significant extension of our previous theorem. The processes under consideration here are transient; but further extension to the construction of recurrent processes (recurrent in the sense of oneand two-dimensional Brownian motions) with smooth hitting distributions does not seem too difficult, using the theorem of [2] mentioned above.