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Local time and Tanaka formulae for super Brownian and super stable processes

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  • Adler, Robert J.
  • Lewin, Marica

Abstract

We develop Tanaka-like evolution equations describing the local time Lxt of certain measure valued super processes. For example, if Xt, t [greater-or-equal, slanted] 0, is a planar super Brownian motion and [lambda] > 0 then L = - + [lambda] [integral operator]t0 ds + [integral operator]t0 , where X[Delta] is a martingale measure associated with X, and G[lambda] is the Green's function of a planar Brownian motion (B1t, B2t) standardised to have E[B(i)1]2 = 2. Properties such as the continuity of Lxt in t and x are immediate consequences of these results. En passant, we also establish that Brownian and stable super processes (in the appropriate dimensions) integrate p functions, and derive an Itô formula for these processes more general than others derived previously.

Suggested Citation

  • Adler, Robert J. & Lewin, Marica, 1992. "Local time and Tanaka formulae for super Brownian and super stable processes," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 45-67, May.
    • Handle: RePEc:eee:spapps:v:41:y:1992:i:1:p:45-67
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    Cited by:

    1. Ethier, S. N. & Krone, Stephen M., 1995. "Comparing Fleming-Viot and Dawson-Watanabe processes," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 171-190, December.
    2. Kaj, I. & Salminen, P., 1995. "On the ultimate value of local time of one-dimensional super-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 21-42, September.
    3. Bojdecki, Tomasz & Talarczyk, Anna, 2005. "Particle picture approach to the self-intersection local time of density processes in," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 449-479, March.
    4. Krone, Stephen M., 1997. "Representations for continuous additive functionals of super-Brownian and super-stable processes," Statistics & Probability Letters, Elsevier, vol. 34(3), pages 211-223, June.
    5. L. Mytnik & K.-N. Xiang, 2004. "Tanaka Formulae for (α, d, β)-Superprocesses," Journal of Theoretical Probability, Springer, vol. 17(2), pages 483-502, April.
    6. Dawson, D.A. & Vaillancourt, J. & Wang, H., 2021. "Joint Hölder continuity of local time for a class of interacting branching measure-valued diffusions," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 212-233.
    7. Hambly, Ben & Koepernik, Peter, 2023. "Dimension results and local times for superdiffusions on fractals," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 377-417.
    8. Bojdecki, Tomasz & Gorostiza, Luis G., 1995. "Self-intersection local time for Gaussian '(d)-processes: Existence, path continuity and examples," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 191-226, December.
    9. Feldman, Raisa E. & Iyer, Srikanth K., 1996. "A representation for functionals of superprocesses via particle picture," Stochastic Processes and their Applications, Elsevier, vol. 64(2), pages 173-186, November.

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