Large-deviation bounds for cluster coagulation models with gelation

RESUMO

We establish pathwise large-deviation bounds for the empirical-measure flow of a cluster coagulation model with gelling kernels on a locally compact Polish state space. The cluster coagulation process is a measure-valued Markov jump process in which pairs of clusters merge according to a time-inhomogeneous kernel $K(t,x,y,d z)$. Kernels with sufficiently rapid growth may produce gelation: a phase transition in which mass escapes from the finite-cluster population in finite time. We first prove tightness of the empirical flows and identify every subsequential limit as a solution of the time-inhomogeneous multi-type Flory equation, which accounts for post-gelation dynamics through a conserved quantity. We then establish a global large-deviation upper bound on path space with a variational rate function. We also prove a local lower bound near paths that are unique solutions of suitable tilted Flory equations. Finally, we give a convex-dual representation of the upper rate function, including the possible singular part of the representing measure. Our results apply to locally compact Polish state spaces with continuous mass functions, encompassing both the gelling and non-gelling regimes within a unified framework.