Electronic Communications in Probability

From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth

Etienne Pardoux and Anton Wakolbinger

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We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. In Le et al. (2011) such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams (1988).

Article information

Electron. Commun. Probab., Volume 16 (2011), paper no. 63, 720-731.

Accepted: 20 November 2011
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
Secondary: 60J55: Local time and additive functionals 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H10: Stochastic ordinary differential equations [See also 34F05]

Ray-Knight representation local time Feller branching with logistic growth Brownian motion local time drift Girsanov transform

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Pardoux, Etienne; Wakolbinger, Anton. From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth. Electron. Commun. Probab. 16 (2011), paper no. 63, 720--731. doi:10.1214/ECP.v16-1679. https://projecteuclid.org/euclid.ecp/1465262019

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  • Aldous, David. The continuum random tree. I. Ann. Probab. 19 (1991), no. 1, 1–28.
  • Delmas, J.-F. Height process for super-critical continuous state branching process. Markov Process. Related Fields 14 (2008), no. 2, 309–326.
  • Friedman, Avner. Stochastic differential equations and applications. Vol. 1. Probability and Mathematical Statistics, Vol. 28. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. xiii+231 pp.
  • Lambert, Amaury. The branching process with logistic growth. Ann. Appl. Probab. 15 (2005), no. 2, 1506–1535.
  • V. Le, E. Pardoux, A. Wakolbinger,”Trees under attack”: a Ray-Knight representation of Feller's branching diffusion with logistic growth, http://www.cmi.univ-.fr/~pardoux/LPW-11.pdf, to appear in Probab. Th. Rel. Fields
  • Le Gall, Jean-François. Itô's excursion theory and random trees. Stochastic Process. Appl. 120 (2010), no. 5, 721–749.
  • Norris, J. R.; Rogers, L. C. G.; Williams, David. Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Related Fields 74 (1987), no. 2, 271–287.
  • E. Pardoux, A Wakolbinger, From exploration paths to mass excursions - variations on a theme of Ray and Knight, in: Surveys in Stochastic Processes, Proceedings of the 33rd SPA Conference in Berlin, 2009, J. Blath, P. Imkeller, S. Roelly (eds.), pp. 87–106, EMS 2011.
  • Pitman, Jim. The distribution of local times of a Brownian bridge. Séminaire de Probabilités, XXXIII, 388–394, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
  • Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 2. Itô calculus. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xiv+480 pp. ISBN: 0-521-77593-0