Convergence to equilibrium of critical branching particle systems and superprocesses, and related nonlinear partial differential equations (Q1200446)

It is well-known that branching \(\alpha\)-stable processes in \(R^ d\) with spatially homogeneous ``\(\beta\)-stable'' branching are persistent and have finite intensity spatially homogeneous invariant measures if and only if \(d>d_ 0=\alpha/\beta\) and a similar result holds for the corresponding superprocesses. The present paper is concerned with some generalizations and extensions of this phenomenon. In particular it is proved that for large classes of initial states the particle process and superprocess converge in distribution towards an equilibrium state as time tends to infinity. Moreover the results include multitype branching systems with a finite number of types. Exploiting the relation between the particle systems and superprocesses and the corresponding system of nonlinear partial differential equations (which arise as the ``log- Laplace equations'') they obtain results on the asymptotic behavior of the latter.