The Annals of Probability

The front location in branching Brownian motion with decay of mass

Louigi Addario-Berry and Sarah Penington

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles’ masses decay. In standard BBM, we may define the front displacement at time $t$ as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from $\Theta(1)$ to $o(1)$. We show that one can find arbitrarily large times $t$ for which this occurs at a distance $\Theta(t^{1/3})$ behind the front displacement for standard BBM.

Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3752-3794.

Dates
Received: December 2015
Revised: August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1511773663

Digital Object Identifier
doi:10.1214/16-AOP1148

Mathematical Reviews number (MathSciNet)
MR3729614

Zentralblatt MATH identifier
06838106

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J65: Brownian motion [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 82C22: Interacting particle systems [See also 60K35]

Keywords
Branching Brownian motion branching interacting particle systems front propagation consistent maximal displacement Brownian motion

Citation

Addario-Berry, Louigi; Penington, Sarah. The front location in branching Brownian motion with decay of mass. Ann. Probab. 45 (2017), no. 6A, 3752--3794. doi:10.1214/16-AOP1148. https://projecteuclid.org/euclid.aop/1511773663


Export citation

References

  • [1] Addario-Berry, L., Berestycki, N. and Gantert, N. (2013). Extremes in branching random walk and branching Brownian motion. Oberwolfach Rep. 10 1205–1251.
  • [2] Berestycki, H., Nadin, G., Perthame, B. and Ryzhik, L. (2009). The non-local Fisher-KPP equation: Travelling waves and steady states. Nonlinearity 22 2813–2844.
  • [3] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford.
  • [4] Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531–581.
  • [5] Britton, N. F. (1989). Aggregation and the competitive exclusion principle. J. Theoret. Biol. 136 57–66.
  • [6] Britton, N. F. (1990). Spatial structures and periodic travelling waves in an integro-differential reaction–diffusion population model. SIAM J. Appl. Math. 50 1663–1688.
  • [7] Etheridge, A. M. (2004). Survival and extinction in a locally regulated population. Ann. Appl. Probab. 14 188–214.
  • [8] Gourley, S. A. (2000). Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol. 41 272–284.
  • [9] Harris, S. and Roberts, M. I. (2011). The many-to-few lemma and multiple spines. Available at arXiv:1106.4761 [math.PR].
  • [10] Harris, S. C., Hesse, M. and Kyprianou, A. E. (2016). Branching Brownian motion in a strip: Survival near criticality. Ann. Probab. 44 235–275.
  • [11] Hu, Y. and Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 742–789.
  • [12] Jaffuel, B. (2012). The critical barrier for the survival of branching random walk with absorption. Ann. Inst. Henri Poincaré Probab. Stat. 48 989–1009.
  • [13] Lambert, A. (2005). The branching process with logistic growth. Ann. Appl. Probab. 15 1506–1535.
  • [14] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin. 16 195–248. Springer, Berlin.
  • [15] McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28 323–331.
  • [16] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Univ. Press, Cambridge.
  • [17] Roberts, M. I. (2015). Fine asymptotics for the consistent maximal displacement of branching Brownian motion. Electron. J. Probab. 20 1–26.
  • [18] Volpert, V. and Petrovskii, S. (2009). Reaction–diffusion waves in biology. Physics of Life Reviews 6 267–310.
  • [19] Watanabe, S. (1965). On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto Univ. 4 385–398.