Acta Applicandae Mathematicae

, Volume 134, Issue 1, pp 201–228 | Cite as

Branching Brownian Motion with Catalytic Branching at the Origin

  • Sergey Bocharov
  • Simon C. Harris


We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate β>0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above λt at time t, for λ>0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.


Catalytic branching Brownian motion 


  1. 1.
    Aïdékon, E., Berestycki, J., Brunet, É., Shi, Z.: The branching Brownian motion seen from its tip. Probab. Theory Relat. Fields 157, 405–451 (2013) CrossRefzbMATHGoogle Scholar
  2. 2.
    Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Probab. 20, 137–151 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel (2002) CrossRefzbMATHGoogle Scholar
  4. 4.
    Carmona, P., Hu, Y.: The spread of a catalytic branching random walk. Ann. Inst. Henri Poincaré Probab. Stat. 50(2), 327–351 (2014) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Dawson, D.A., Fleischmann, K.: A super-Brownian motion with a single point catalyst. Stoch. Process. Appl. 49(1), 3–40 (1994) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Döring, L., Roberts, M.: Catalytic branching processes via spine techniques and renewal theory. Sém. Probab. XLV, 305–322 (2013) Google Scholar
  7. 7.
    Durrett, R.: Probability: Theory and Examples, 2nd edn. Duxbury, N. Scituate (1996) Google Scholar
  8. 8.
    Engländer, J., Turaev, D.: A scaling limit theorem for a class of superdiffusions. Ann. Probab. 30(2), 683–722 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Engländer, J., Harris, S.C., Kyprianou, A.E.: Strong law of large numbers for branching diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 46(1), 279–298 (2010) CrossRefzbMATHGoogle Scholar
  10. 10.
    Fleischmann, K., Le Gall, J.-F.: A new approach to the single point catalytic super-Brownian motion. Probab. Theory Relat. Fields 102(1), 63–82 (1995) CrossRefzbMATHGoogle Scholar
  11. 11.
    Hardy, R., Harris, S.C.: A spine approach to branching diffusions with applications to Lp-convergence of martingales. PhD thesis, University of Cambridge (1995). Also in Séminaire de Probabilités, XLII, pp. 281–330 (2009) Google Scholar
  12. 12.
    Harris, J.W., Harris, S.C.: Branching Brownian motion with an inhomogeneous breeding potential. Ann. Inst. Henri Poincaré Probab. Stat. 45(3), 793–801 (2009) CrossRefzbMATHGoogle Scholar
  13. 13.
    Karatzas, I., Shreve, S.E.: Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. Ann. Probab. 12, 819–828 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1994) zbMATHGoogle Scholar
  15. 15.
    Roberts, M.: A simple path to asymptotics for the frontier of a branching Brownian motion. Ann. Probab. 41(5), 3518–3541 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Rogers, L.G.C.: A guided tour through excursions. Bull. Lond. Math. Soc. 21(4), 305–341 (1989) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations