Stochastic flows and Poisson representations for the block masses of the $\Lambda$-coalescent with dust : moment asymptotics and large deviation estimates

Abstract

We develop a new methodology for the study of the $\Lambda$-coalescent with dust, based on the construction of a stochastic flow of inverses, introduced by Bertoin and Le Gall [Ann. inst. Henri Poincare (B) Probab. Stat. 41(3), 307-333 (2003)], in which the coalescent is naturally embedded as a nested interval-partition. This framework yields Poisson representations for the ordered block masses $(W_k(t))_{k \geq 1}$ as stochastic integrals with respect to the Poisson random measure governing the flow, enabling the use of stochastic calculus in a setting where it was not previously available. We believe this methodology to be of independent interest and applicable beyond the specific results of this paper. As a first application, we derive precise logarithmic asymptotics for the moments of $W_k(t)$ as $t \to \infty$, which reveal an interesting cutoff phenomenon related to the presence of dust. We then establish a law of large numbers and a large deviation principle for $\log(1-W_1(t))/t$ and a one-sided weak large deviation principle for $\log(W_k(t))/t$, for $k \geq 2$, with explicit rate functions.