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Abstract
An important property of Kingman’s coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as “coming down from infinity”. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingman’s coalescent is the “fastest” to come down from infinity. In this article, we study what happens when we counteract this “fastest” coalescent with the action of an extreme form of fragmentation. We augment Kingman’s coalescent, where any two blocks merge at rate c>0, with a fragmentation mechanism where each block fragments at constant rate, λ>0, into its constituent elements. We prove that there exists a phase transition at λ=c/2, between regimes where the resulting “fast” fragmentationcoalescence process is able to come down from infinity or not. In the case that λ<c/2, we develop an excursion theory for the fast fragmentationcoalescence process out of which a number of interesting quantities can be computed explicitly.
Original language  English 

Pages (fromto)  38293849 
Number of pages  21 
Journal  Annals of Probability 
Volume  45 
Issue number  6A 
Early online date  27 Nov 2017 
DOIs  
Publication status  Published  30 Nov 2017 
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 1 Finished

RealValued SelfSimilar Markov Processes and their Applications
Kyprianou, A. (PI)
Engineering and Physical Sciences Research Council
2/06/14 → 1/10/17
Project: Research council