Electronic Journal of Probability

Williams decomposition for superprocesses

Yan-Xia Ren, Renming Song, and Rui Zhang

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We decompose the genealogy of a general superprocess with spatially dependent branching mechanism with respect to the last individual alive (Williams decomposition). This is a generalization of the main result of Delmas and Hénard [5] where only superprocesses with spatially dependent quadratic branching mechanism were considered. As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total measure will converge to a point measure at its extinction time. This partially generalizes a result of Tribe [27] in the sense that our branching mechanism is more general.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 23, 33 pp.

Received: 12 September 2016
Accepted: 29 January 2018
First available in Project Euclid: 27 February 2018

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Digital Object Identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60G55: Point processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

superprocesses Williams decomposition spatially dependent branching mechanism genealogy

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Ren, Yan-Xia; Song, Renming; Zhang, Rui. Williams decomposition for superprocesses. Electron. J. Probab. 23 (2018), paper no. 23, 33 pp. doi:10.1214/18-EJP146. https://projecteuclid.org/euclid.ejp/1519722152

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  • [1] Abraham, R. and Delmas, J.-F.: Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119, (2009), 1124–1143.
  • [2] Chen, Z.-Q., Ren, Y.-X. and Wang, H.: An almost sure scaling limit theorem for Dawson-Watanabe superprocesses. J. Funct. Anal. 254, (2008), 1988–2019.
  • [3] Chen, Z.-Q. and Zhang, X.: Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Related Fields 165, (2016), 267–312.
  • [4] Dawson, D. A.: Measure-Valued Markov Processes. École d’Été de Probabilités de Saint-Flour XXI–1991, 1–260, Lecture Notes in Math., 1541. Springer, Berlin, 1993.
  • [5] Delmas, J. F. and Hénard, O.: A Williams decomposition for spatially dependent super-processes. Electron. J. Probab. 18, (2013), No. 14, 1–43.
  • [6] Dynkin, E. B.: Superprocesses and partial differential equations. Ann. Probab. 21, (1993), 1185–1262.
  • [7] Dynkin, E. B. and Kuznetsov, S. E.: $\mathbb N$-measure for branching exit Markov system and their applications to differential equations. Probab. Theory Related Fields 130, (2004), 135–150.
  • [8] Eckhoff, M., Kyprianou, A. E. and Winkel M.: Spines, skeletons and the strong law of large numbers for superdiffusions. Ann. Probab. 43, (2015), 2545–2610.
  • [9] Englander, J., Ren, Y.-X. and Song, R.: Weak extinction versus global exponential growth of total mass for superdiffusions. Ann. Inst. Henri Poincaré Probab. Stat. 52, (2016), 448–482.
  • [10] Garroni, M. G. and Menaldi, J.-L.: Green functions for second order parabolic integro-differential problems. Pitman Research Notes in Mathematics Series, 275. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992,
  • [11] Grey, D. R.: Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11, (1974), 669–677.
  • [12] El Karoui, N. and Roelly, S.: Propriétés de martingales, explosion et représentation de Lévy-Khintchine d’une classe de processus de branchment à valeurs mesures. Stoch. Proc. Appl. 38, (1991), 239–266.
  • [13] Kim, K., Song, R. and Vondracek, Z.: Heat kernels of non-symmetric jump processes: beyond the stable case. Potential Anal., (2017). https://doi.org/10.1007/s11118-017-9648-4
  • [14] Kyprianou, A. E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer-Verlag, Berlin, 2006.
  • [15] Kyprianou, A. E., Liu, R.-L., Murillo-Salas, A. and Ren, Y.-X.: Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Ann. Inst. Henri Poincaré Probab. Stat. 48, (2012), 661–687.
  • [16] Kyprianou, A. E., Pérez, J.-L., and Ren, Y.-X.: The backbone decomposition for spatially dependent supercritical superprocesses. Séminaire de Probabilités XLVI, 33–59, Lecture Notes in Math., 2123, Springer, Cham, 2014.
  • [17] Ladyzenskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N.: Linear and Quasi-linear Equations of Parabolic Type. American Math. Soc., Providence, Rhode Island, 1968.
  • [18] Li, Z.: Skew convolution semigroups and related immigration processes. Theory Probab. Appl. 46, (2003), 274–296.
  • [19] Li, Z.: Measure-Valued Branching Markov Processes. Springer, Heidelberg, 2011.
  • [20] Liu, R.-L., Ren, Y.-X. and Song R.: $L\log L$ criterion for a class of superdiffusions. J. Appl. Probab. 46, (2009), 479–496.
  • [21] Liu, R.-L., Ren, Y.-X. and Song, R.: Strong law of large numbers for a class of superdiffusions. Acta Appl. Math. 123, (2013), 73–97.
  • [22] Perkins, E.: Dawson-Watanable superprocesses and measure-valued diffusions. Lectures on Probability Theory and Statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781. Springer-Verlag, Heidelberg, 2002, 135–192.
  • [23] Ren, Y.-X., Song, R. and Zhang, R.: Limit theorems for some critical superprocesses. Illinois J. Math. 59, (2015), 235–276.
  • [24] Ren, Y.-X., Song, R. and Yang, T.: Spine decomposition and $LlogL$ criterion for superprocesses with non-local branching mechanisms. arXiv:1609.02257 [math.PR]
  • [25] Sheu, Y.-C.: Lifetime and compactness of range for super-Brownian motion with a general branching mechanism. processes. Stochastic Process. Appl. 70, (1997), 129–141.
  • [26] Stroock, D. W.: Probability Theory. An Analytic View. 2nd ed. Cambridge University Press, Cambridge, 2011.
  • [27] Tribe, R.: The behavior of superprocesses near extinction. Ann. Probab. 20, (1992), 286-311.