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 X_s (ϕ ϵ)ds, where {ϕ_ϵ} is a sequence of smooth functions converging to δ₀ in distributional sense as ϵ → 0. We prove that the limit L_t ⁰ exists if d < (1 + 1/β)α and does not depend of the particular choice of sequence {ϕ_ϵ}. We show that L_t 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.

Keywords:

• Infinite variance superprocesses
• Local time
• Tanaka like formula

2000 Mathematics Subject Classification:

• Primary 60J55
• Secondary 60G57

Acknowledgments

This work is part of my Ph.D. Thesis, Villa (2002) Representaciones tipo Fórmula de Tanaka del Tiempo Local de Superprocesos, Ph.D. Thesis, CIMAT, written under the supervision of Professor J. A. López-Mimbela. I would like to thank an anonymous referee for his helpful remarks.