Ergodic theory and a local occupation time for measure-valued critical branching brownian motion

Abstract

Let (Xt)t>=0 denote the measure-valued critical branching Brownian motion on Rd with initial state being Lebesgue measure. A strong ergodic theorem is proved for (Xt)t>=0 when d>=3, while a weak ergodic theorem is proved for d = 2. Also a weak local occupation time (an analogue of the local time for Brownian motion) is shown to exist in dimensions d=1,2 and 3.