Branching one-dimensional periodic diffusion processes

Let X be a nonsingular conservative one-dimensional periodic diffusion process, [lambda]0 its principal eigenvalue and X a binary splitting branching diffusion process with nonbranching part X. For each [alpha] > [lambda]0 we have two positive martingales Wit([alpha]), i = 1, 2, of X attached to the two positive [alpha]-harmonic functions of X. The main purpose of this paper is to show that their limit random variables are positive for all [alpha] [epsilon] ([lambda]0, [alpha]i), where [alpha]i are some constants greater than [lambda]0.