Stochastic Coalescence Multi-Fragmentation Processes

Abstract: We study infinite systems of particles which undergo coalescence and fragmentation, in a manner determined solely by their masses. A pair of particles having masses $x$ and $y$ coalesces at a given rate $K(x,y)$. A particle of mass $x$ fragments into a collection of particles of masses $\theta_1 x, \theta_2 x, \ldots$ at rate $F(x) \beta(d\theta)$. We assume that the kernels $K$ and $F$ satisfy H\"older regularity conditions with indices $\lambda \in (0,1]$ and $\alpha \in [0, \infty)$ respectively. We show existence of such infinite particle systems as strong Markov processes taking values in $\ell_{\lambda}$, the set of ordered sequences $(m_i)_{i \ge 1}$ such that $\sum_{i \ge 1} m_i^{\lambda} \textless{} \infty$. We show that these processes possess the Feller property. This work relies on the use of a Wasserstein-type distance, which has proved to be particularly well-adapted to coalescence phenomena.