Electronic Journal of Probability

A Williams decomposition for spatially dependent superprocesses

Jean-François Delmas and Olivier Hénard

Full-text: Open access


We present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess introduced by Engländer and Pinsky and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (Williams' decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diffusion (investigated by Champagnat and Roelly) and the superdiffusion are provided.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 37, 43 pp.

Accepted: 12 March 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J55: Local time and additive functionals 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Spatially dependent superprocess Williams' decomposition genealogy h-transform Q-process

This work is licensed under a Creative Commons Attribution 3.0 License.


Delmas, Jean-François; Hénard, Olivier. A Williams decomposition for spatially dependent superprocesses. Electron. J. Probab. 18 (2013), paper no. 37, 43 pp. doi:10.1214/EJP.v18-1801. https://projecteuclid.org/euclid.ejp/1465064262

Export citation


  • R. Abraham and J.-F. Delmas. A continuum-tree-valued Markov process. To appear in The Annals of Probability, 2012.
  • Abraham, Romain; Delmas, Jean-François. Williams' decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119 (2009), no. 4, 1124–1143.
  • Aldous, David. The continuum random tree. I. Ann. Probab. 19 (1991), no. 1, 1–28.
  • Bertoin, Jean; Fontbona, Joaquin; Martínez, Servet. On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 (2008), no. 3, 714–726.
  • Bertoin, Jean; Le Gall, Jean-François. The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2000), no. 2, 249–266.
  • Champagnat, Nicolas; Roelly, Sylvie. Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions. Electron. J. Probab. 13 (2008), no. 25, 777–810.
  • Y.-T. Chen and J.-F. Delmas. Smaller population size at the CA time for stationary branching processes. To appear in The Annals of Probability, 2012.
  • Cranston, M.; Koralov, L.; Molchanov, S.; Vainberg, B. A solvable model for homopolymers and self-similarity near the critical point. Random Oper. Stoch. Equ. 18 (2010), no. 1, 73–95.
  • Dhersin, Jean-Stéphane; Serlet, Laurent. A stochastic calculus approach for the Brownian snake. Canad. J. Math. 52 (2000), no. 1, 92–118.
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166–205.
  • Duquesne, Thomas; Le Gall, Jean-François. Random trees, Lévy processes and spatial branching processes. Astérisque No. 281 (2002), vi+147 pp.
  • Duquesne, Thomas; Winkel, Matthias. Growth of Lévy trees. Probab. Theory Related Fields 139 (2007), no. 3-4, 313–371.
  • Dynkin, Eugene B. An introduction to branching measure-valued processes. CRM Monograph Series, 6. American Mathematical Society, Providence, RI, 1994. x+134 pp. ISBN: 0-8218-0269-0
  • Engländer, János; Harris, Simon C.; Kyprianou, Andreas E. Strong law of large numbers for branching diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 279–298.
  • Engländer, János; Kyprianou, Andreas E. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2004), no. 1A, 78–99.
  • Engländer, János; Pinsky, Ross G. On the construction and support properties of measure-valued diffusions on $D\subseteq{\bf R}^ d$ with spatially dependent branching. Ann. Probab. 27 (1999), no. 2, 684–730.
  • Klebaner, F. C.; Rösler, U.; Sagitov, S. Transformations of Galton-Watson processes and linear fractional reproduction. Adv. in Appl. Probab. 39 (2007), no. 4, 1036–1053.
  • Kurtz, Thomas G.; Rodrigues, Eliane R. Poisson representations of branching Markov and measure-valued branching processes. Ann. Probab. 39 (2011), no. 3, 939–984.
  • Le Gall, J.-F. A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 (1993), no. 1, 25–46.
  • Li, Zeng Hu. A note on the multitype measure branching process. Adv. in Appl. Probab. 24 (1992), no. 2, 496–498.
  • Perkins, Edwin. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781, Springer, Berlin, 2002.
  • Pinsky, Ross G. Positive harmonic functions and diffusion. Cambridge Studies in Advanced Mathematics, 45. Cambridge University Press, Cambridge, 1995. xvi+474 pp. ISBN: 0-521-47014-5
  • Roynette, Bernard; Yor, Marc. Penalising Brownian paths. Lecture Notes in Mathematics, 1969. Springer-Verlag, Berlin, 2009. xiv+275 pp. ISBN: 978-3-540-89698-2
  • Seneta, E. Non-negative matrices and Markov chains. Revised reprint of the second (1981) edition [Springer-Verlag, New York; ]. Springer Series in Statistics. Springer, New York, 2006. xvi+287 pp. ISBN: 978-0387-29765-1; 0-387-29765-0
  • Serlet, Laurent. The occupation measure of super-Brownian motion conditioned to nonextinction. J. Theoret. Probab. 9 (1996), no. 3, 561–578.
  • Williams, David. Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3) 28 (1974), 738–768.