Stochastic flows associated to coalescent processes
type: Item
instance of: scholarly article
title: Stochastic flows associated to coalescent processes
zbMATH Open document ID: 1023.92018
author: J.-F. Le Gall
author: Jean Bertoin
publication date: 2003-08-20T00:00:00Z
review text: Define a bridge to be a right-continuous process \((B(r)\), \(r\in [0,1])\) with non-decreasing paths and exchangeable increments, such that \(B(0)=0\) and \(B(1)=1\). The authors show that flows of bridges are in one-to-one correspondence with certain coalescent processes which arise as limits of Cannings' model for genealogy when the size \(N\) of the population tends to infinity. The latter processes, called exchangeable \(\mathcal P\)-coalescents, were studied by \textit{M. Möhle} and \textit{S. Sagitov} [Ann. Probab. 29, 1547-1562 (2001; Zbl 1013.92029)], \textit{J. Pitman} [Ann. Probab. 27, 1870-1902 (1999; Zbl 0963.60079)], \textit{S. Sagitov} [J. Appl. Probab 36, 1116-1902 (1999; Zbl 0962.92026)] and \textit{J. Schweinsberg} [Electron. J. Probab. 5, paper No.~12, (2000; Zbl 0959.60065)]. Flows of bridges associated to the special class of coalescents with multiple collisions (\(\Lambda\)-coalescents) are obtained as a weak limit of simple flows of bridges, that are flows constructed in a certain way from elementary bridges with exactly one jump. The authors discuss an important measure valued process which is closely related to the genealogical structure coded by the coalescent and which can be seen as a generalized Fleming-Viot process.
Mathematics Subject Classification ID: 92D15
Mathematics Subject Classification ID: 60G09
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 1968522
zbMATH Keywords: coalescent
zbMATH Keywords: branching processes
zbMATH Keywords: genealogical structure
zbMATH Keywords: generalized Fleming-Viot process
zbMATH Keywords: exchangeability
zbMATH Keywords: bridge
zbMATH Keywords: stochastic flows
MaRDI profile type: Publication
full work available at URL: s00440-003-0264-4
OpenAlex ID: W2067409890
DOI: 10.1007/S00440-003-0264-4
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1968522