Local Time and a Tanaka Formula for a Class of Discontinuous Measure-Valued Branching Processes

Abstract

We define the local time of a discontinuous superprocess X in ℝ d with symmetric α-stable motions and (1 + β)-branching as an L 1-limit of approximating local times ∫0 t Xs (ϕ ϵ)ds, where {ϕϵ} is a sequence of smooth functions converging to δ0 in distributional sense as ϵ → 0. We prove that the limit Lt 0 exists if d < (1 + 1/β)α and does not depend of the particular choice of sequence {ϕϵ}. We show that Lt 0 admits a Tanaka formula-like representation which includes a term that incorporates the discontinuities of X. Fleischmann [10] proved that the occupation measure induced by X is absolutely continuous with respect to Lebesgue measure; using the Tanaka formula we give an easy proof of Fleischmann's result.