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Stochastic flows associated to coalescent processes. (English) Zbl 1023.92018

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 M. Möhle and S. Sagitov [Ann. Probab. 29, 1547-1562 (2001; Zbl 1013.92029)], J. Pitman [Ann. Probab. 27, 1870-1902 (1999; Zbl 0963.60079)], S. Sagitov [J. Appl. Probab 36, 1116-1902 (1999; Zbl 0962.92026)] and 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.

MSC:

92D15 Problems related to evolution
60G09 Exchangeability for stochastic processes
60J25 Continuous-time Markov processes on general state spaces
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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