The branching random field

Abstract

The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.