Electronic Communications in Probability

Limiting distribution of the rightmost particle in catalytic branching Brownian motion

Sergey Bocharov and Simon C. Harris

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We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $\beta \delta _0(\cdot )$, where $\delta _0(\cdot )$ is the Dirac delta function and $\beta $ is some positive constant. We show that the distribution of the rightmost particle centred about $\frac{\beta } {2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [10] for the degenerate case of catalytic branching.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 70, 12 pp.

Received: 12 May 2016
Accepted: 7 September 2016
First available in Project Euclid: 4 October 2016

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Zentralblatt MATH identifier

Primary: 60J55: Local time and additive functionals 60J65: Brownian motion [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Brownian motion local time catalytic branching

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Bocharov, Sergey; Harris, Simon C. Limiting distribution of the rightmost particle in catalytic branching Brownian motion. Electron. Commun. Probab. 21 (2016), paper no. 70, 12 pp. doi:10.1214/16-ECP22. https://projecteuclid.org/euclid.ecp/1475601101

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  • [1] Bocharov, S. and Harris S. C.: Branching Brownian Motion with catalytic branching at the origin. Acta Applicandae Mathematicae 134, (2014), 201–228.
  • [2] Bramson, M., Ding, J. and Zeitouni, O.: Convergence in law of the maximum of nonlattice branching random walk, arXiv:1404.3423v1
  • [3] Dawson, D. A. and Fleischmann, K.: A Super-Brownian motion with a single point catalyst. Stochastic Processes and their Applications 49(1), (1994), 3–40.
  • [4] Engländer, J. and Turaev, D.: A scaling limit theorem for a class of superdiffusions. The Annals of Probability 30(2), (2002), 683–722.
  • [5] Fleischmann, K. and Le Gall, J-F.: A new approach to the single point catalytic super-Brownian motion. Probability Theory and Related Fields 102(1), (1995), 63–82.
  • [6] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion. (3rd ed.) Springer, (1999).
  • [7] Harris, S. C. and Roberts, M.: The many-to-few lemma and multiple spines, arXiv:1106.4761
  • [8] Karatzas, I. and Shreve, S. E.: Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. The Annals of Probability 12(3), (1984), 819–828.
  • [9] Lalley, S. and Sellke, T.: A conditional limit theorem for the frontier of a branching Brownian motion. The Annals of Probability 15(3), (1987), 1052–1061.
  • [10] Lalley, S. and Sellke, T.: Travelling waves in inhomogeneous branching Brownian motions. I. The Annals of Probability 16(3), (1988), 1051–1062. MR0942755