Historical Lattice Trees

Abstract

We prove that the rescaled historical processes associated to critical spread-out lattice trees in dimensions d> 8 converge to historical Brownian motion. This is a functional limit theorem for measure-valued processes that encodes the genealogical structure of the underlying random trees. Our results are applied elsewhere to prove that random walks on lattice trees, appropriately rescaled, converge to Brownian motion on super-Brownian motion.