Branching one-dimensional periodic diffusion processes
type: Item
instance of: scholarly article
title: Branching one-dimensional periodic diffusion processes
zbMATH Open document ID: 0557.60073
DOI: 10.1016/0304-4149(85)90040-7
author: Nobuyuki Ikeda
author: Kiyoshi Kawazu
author: Yukio Ogura
published in: Stochastic Processes and their Applications
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 3888663
zbMATH Keywords: splitting branching diffusion process
zbMATH Keywords: periodic, conservative diffusion
zbMATH Keywords: principal eigenvalue
zbMATH Keywords: martingale theory
reviewed by: Paavo Salminen
MaRDI profile type: Publication
full work available at URL: 0304-4149(85)90040-7
OpenAlex ID: W1993593689
cites work: Q4189539
cites work: Maximal displacement of branching brownian motion
cites work: Q5848416
cites work: Q5847356
cites work: The parabolic differential equations and the associated semigroups of transformation
cites work: Q4086525
cites work: Q3855442
cites work: Branching Markov processes. III
cites work: Q5332526
cites work: On a limit theorem for branching one-dimensional diffusion processes
cites work: Branching Brownian motion with absorption
cites work: Q5768532
cites work: Q3959913
cites work: Q5540675
cites work: Elementary Solutions for Certain Parabolic Partial Differential Equations
cites work: Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov
cites work: Q3492541
cites work: Spatial growth of a branching process of particles living in \(R^ n\).
cites work: Q5667341
description: scientific article; zbMATH DE number 3888663