Branching one-dimensional periodic diffusion processes

Abstract

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