Clumping in multitype-branching trees

The authors investigate the ‘clumping versus local finiteness’ behavior in the infinite backward tree for a class of branching particle systems in ℝ’d with symmetric stable migration and critical ‘genuine multitype’ branching. Under mild assumptions on the branching they establish, by analysing certain ergodic properties of the individual ancestral process, a critical dimension dc such that the (measure-valued) tree-top is almost surely locally finite if and only if d>dc. This result is used to obtain L¹-norm asymptotics of a corresponding class of systems of non-linear partial differential equations.