On Parabolic Functions of One-Dimensional Quasidiffusions

Abstract

In this paper we will deal with quasidiffusions X=(Xt)t^0 on [0, 1) assuming that 0 is a reflecting regular boundary, 1 is an accessible or entrance boundary and X is killed as soon as it hits the boundary 1. We shall give a Martin representation of space-time-excessive functions for X. This includes in particular a representation of all parabolic functions f(x, t) satisfying a certain integrability condition by minimal ones (see Theorem 2 below). We shall show that the set of minimal points of space-time Martin boundary is homeomorphic to (0, oo[; in particular the minimal parabolic functions form a one-parameter family (kt) (re(0, oo]). If z<oo5 the function kt(x,s) is the limit (in a weak sense) of the transition density p(t—s, x, y) or its derivative with respect to j, where y converges to the boundary 1. In the limit circle case we will give an uniformly convergent expansion of kt (t^(Q, °o]) in eigenfunctions of the infinitesimal operator of X. Using these results we consider the problem which minimal parabolic functions factorize (i.e. have the form kt(x, s) = 0*(XhMX)) and which factorizing parabolic functions are minimal. (For some Markov chains and diffusion processes this problem was studied in [9], [11].) As another application we shall give a necessary and sufficient condition in order that for a parabolic function/the process {f(Xt, s+t), t^[a, b]} is a martingale. Parabolic functions which are martingales on the trajectories of X are used to determine the probabilities that X ever hits some time-varying boundaries (see e.g. [12]) and were studied e.g. in [2], [8]. The assumption that the boundary 1 is accessible or entrance is essential.