A PHASE TRANSITION IN THE COMING DOWN FROM INFINITY OF SIMPLE EXCHANGEABLE FRAGMENTATION-COAGULATION PROCESSES

Clément Foucart (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $\Lambda$-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes, simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters θ ≤ θ ∈ [0, ∞], so that if θ^{\star} < 1, the process comes down from infinity and if θ_{\star} > 1, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters θ^{\star} , θ_{\star} coincide and are explicit.

Mots clés: Coupling; Exchangeability; Partition-valued processes; Boundary behavior; Coming down from infinity; Fragmentation

Domaines: Probabilités [math.PR]

Clément Foucart. A PHASE TRANSITION IN THE COMING DOWN FROM INFINITY OF SIMPLE EXCHANGEABLE FRAGMENTATION-COAGULATION PROCESSES. 2021. ⟨hal-03211894⟩