Branching one-dimensional periodic diffusion processes
Let X be a binary splitting branching diffusion process on \({\mathbb{R}}\). The number of particles alive at time t in a Borel set \(A\subseteq {\mathbb{R}}\) is denoted by \(Z_ t=Z_ t(1_ A;X)\). In the paper under review the asymptotic behaviour as \(t\to \infty\) of \(Z_ t\) is studied under the assumption that the non-branching part of X is a periodic, conservative diffusion. A diffusion X is called periodic with period 1 if the space of all continuous periodic functions with period 1 is invariant under the semigroup of X. It is seen that periodicity implies nice properties for the principal eigenvalue \(\lambda_ 0\geq 0\) and the fundamental solutions \(g_ 1(\alpha;\cdot),g_ 2(\alpha;\cdot)\) \((\alpha \geq - \lambda_ 0)\) associated with the differential operator of X. Combining these results with some martingale theory it is proved that there exists non-negative random variables \(w_ i(\alpha)\), \(i=1,2\), such that \[ \lim_{t\to \infty}e^{-(\alpha +c)t}Z_ t(f;X)=rw^ i(\alpha)\quad with\quad probability\quad 1. \] Here \(\alpha \geq \lambda_ 0\), \(c>0\) is the branching rate, f is a continuous function on \({\mathbb{R}}\) having the growth rate \(r\in {\mathbb{R}}\) when compared with \(g_ i(\alpha;\cdot)\). Further for \(c>\lambda_ 0\) there exists \(\alpha_ i>0\), \(i=1,2\), such that for \(-\lambda_ 0\leq \alpha <\alpha_ i\), \({\mathbb{P}}_ 0(0<w^ i(\alpha)<\infty)=1\). Some examples are also discussed.
scientific article; zbMATH DE number 3888663
scientific article