Annals of Applied Probability

Exponential growth rates in a typed branching diffusion

Y. Git, J. W. Harris, and S. C. Harris

Full-text: Open access


We study the high temperature phase of a family of typed branching diffusions initially studied in [Astérisque 236 (1996) 133–154] and [Lecture Notes in Math. 1729 (2000) 239–256 Springer, Berlin]. The primary aim is to establish some almost-sure limit results for the long-term behavior of this particle system, namely the speed at which the population of particles colonizes both space and type dimensions, as well as the rate at which the population grows within this asymptotic shape. Our approach will include identification of an explicit two-phase mechanism by which particles can build up in sufficient numbers with spatial positions near −γt and type positions near $\kappa \sqrt{t}$ at large times t. The proofs involve the application of a variety of martingale techniques—most importantly a “spine” construction involving a change of measure with an additive martingale. In addition to the model’s intrinsic interest, the methodologies presented contain ideas that will adapt to other branching settings. We also briefly discuss applications to traveling wave solutions of an associated reaction–diffusion equation.

Article information

Ann. Appl. Probab., Volume 17, Number 2 (2007), 609-653.

First available in Project Euclid: 19 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Spatial branching process branching diffusion multi-type branching process additive martingales spine decomposition


Git, Y.; Harris, J. W.; Harris, S. C. Exponential growth rates in a typed branching diffusion. Ann. Appl. Probab. 17 (2007), no. 2, 609--653. doi:10.1214/105051606000000853.

Export citation


  • Athreya, K. B. (2000). Change of measures for Markov chains and the $L\log L$ theorem for branching processes. Bernoulli 6 323–338.
  • Champneys, A., Harris, S. C., Toland, J., Warren, J. and Williams, D. (1995). Algebra, analysis and probability for a coupled system of reaction-diffusion equations. Philos. Trans. Roy. Soc. London 350 69–112.
  • Chauvin, B. and Rouault, A. (1988). KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Related Fields 80 299–314.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Enderle, K. and Hering, H. (1982). Ratio limit theorems for branching Orstein–Uhlenbeck processes. Stochastic Process. Appl. 13 75–85.
  • Engländer, J. and Kyprianou, A. E. (2004). Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 78–99.
  • Geiger, J. (1999). Elementary new proofs of classical limit theorems for Galton–Watson processes. J. Appl. Probab. 36 301–309.
  • Hardy, R. and Harris, S. C. (2006). A new formulation of the spine approach to branching diffusions. Available at
  • Hardy, R. and Harris, S. C. (2006). Spine proofs for $\mathcalL^p$-convergence of branching diffusion martingales. Available at
  • Hardy, R. and Harris, S. C. (2006). A spine proof of a large-deviations principle for branching Brownian motion. Stochastic Process. Appl. 116 1992–2013.
  • Harris, S. C. (1999). Travelling-waves for the FKPP equation via probabilistic arguments. Proc. Roy. Soc. Edinburgh Sect. A 129 503–517.
  • Harris, S. C. (2000). Convergence of a “Gibbs–Boltzmann” random measure for a typed branching diffusion. Séminaire de Probabilités XXXIV. Lecture Notes in Math. 1729 239–256. Springer, Berlin.
  • Harris, S. C. and Williams, D. (1996). Large deviations and martingales for a typed branching diffusion. I. Astérisque 236 133–154.
  • Harris, T. E. (2002). The Theory of Branching Processes. Dover, Mineola, NY.
  • Itô, K. and McKean, H. P. (1965). Diffusion Processes and Their Sample Paths. Academic Press, New York.
  • Kurtz, T., Lyons, R., Pemantle, R. and Peres, Y. (1997). A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) (K. B. Athreya and P. Jagers, eds.). IMA Vol. Math. Appl. 84 181–185. Springer, New York.
  • Kyprianou, A. E. (2004). Travelling wave solutions to the K–P–P equation: Alternatives to Simon Harris' probabilistic analysis. Ann. Inst. H. Poincaré Probab. Statist. 40 53–72.
  • Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) (K. B. Athreya and P. Jagers, eds.). IMA Vol. Math. Appl. 84 217–221. Springer, New York.
  • Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov-Petrovskiĭ–Piskunov. Comm. Pure Appl. Math. 28 323–331.
  • Murray, J. D. (2003). Mathematical Biology. II, 3rd ed. Springer, New York.
  • Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes 1987 (E. Çinlar, K. L. Chung and R. K. Getoor, eds.) 223–242. Birkhäuser, Boston.
  • Olofsson, P. (1998). The $x\log x$ condition for general branching processes. J. Appl. Probab. 35 537–544.
  • Varadhan, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia.
  • Watanbe, S. (1967). Limit theorem for a class of branching processes. In Markov Processes and Potential Theory (J. Chover, ed.) 205–232. Wiley, New York.