Shape Theorems for Poisson Hail on a Bivariate Ground

We consider an extension of the Poisson Hail model where the service speed is either zero or infinity at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process and on the geometrical properties of the zero speed set. 60D05, 60F15, 60G55 Point process theory, Poisson rain, stochastic geometry, random closed set, time and space growth, shape, queuing theory, max-plus algebra, heaps, branching process, sub-additive ergodic theory.