Advertisement

Bessel Processes, the Brownian Snake and Super-Brownian Motion

  • Jean-François Le Gall
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

We prove that, both for the Brownian snake and for super-Brownian motion in dimension one, the historical path corresponding to the minimal spatial position is a Bessel process of dimension − 5. We also discuss a spine decomposition for the Brownian snake conditioned on the minimizing path.

References

  1. 1.
    N. Curien, J.-F. Le Gall, The Brownian plane. J. Theor. Probab. 27(4), 1249–1291 (2014).zbMATHCrossRefGoogle Scholar
  2. 2.
    N. Curien, J.-F. Le Gall, The hull process of the Brownian plane (submitted) [arXiv:1409.4026]Google Scholar
  3. 3.
    D.A. Dawson, I. Iscoe, E.A. Perkins, Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Relat. Fields 83, 135–205 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    D.A. Dawson, E.A. Perkins, Historical processes. Mem. Am. Math. Soc. 93(454), 1–179 (1991)MathSciNetGoogle Scholar
  5. 5.
    J.-F. Delmas, Some properties of the range of super-Brownian motion. Probab. Theory Relat. Fields 114, 505–547 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Duquesne, J.-F. Le Gall, Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque, vol. 281 (Société mathématique de France, Paris, 2002)Google Scholar
  7. 7.
    E.B. Dynkin, Path processes and historical superprocesses. Probab. Theory Relat. Fields 90, 1–36 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    J.-F. Le Gall, Hitting probabilities and potential theory for the Brownian path-valued process. Ann. Inst. Fourier 44, 277–306 (1994)zbMATHCrossRefGoogle Scholar
  9. 9.
    J.-F. Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich (Birkhäuser, Boston, 1999)Google Scholar
  10. 10.
    J.-F. Le Gall, Random trees and applications. Probab. Surv. 2, 245–311 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    J.-F. Le Gall, L. Ménard, Scaling limits for the uniform infinite quadrangulation. Ill. J. Math. 54, 1163–1203 (2010)zbMATHGoogle Scholar
  12. 12.
    J.-F. Le Gall, M. Weill, Conditioned Brownian trees. Ann. Inst. Henri. Poincaré Probab. Stat. 42, 455–489 (2006)zbMATHCrossRefGoogle Scholar
  13. 13.
    J. Pitman, M. Yor, Bessel processes and infinitely divisible laws, in Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Mathematics, vol. 851 (Springer, Berlin, 1981), pp. 285–370Google Scholar
  14. 14.
    J. Pitman, M. Yor, A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59, 425–457 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion (Springer, Berlin, 1991)zbMATHCrossRefGoogle Scholar
  16. 16.
    D. Williams, Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. Lond. Math. Soc. Ser. 3(28), 738–768 (1974)CrossRefGoogle Scholar
  17. 17.
    M. Yor, Loi de l’indice du lacet brownien et distribution de Hartman-Watson. Z. Wahrsch. Verw. Gebiete 53, 71–95 (1980)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité Paris-Sud and Institut universitaire de FranceOrsay CédexFrance

Personalised recommendations