The Annals of Applied Probability

Genealogy of catalytic branching models

Andreas Greven, Lea Popovic, and Anita Winter

Full-text: Open access

Abstract

We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate proportional to the number of catalyst individuals alive. The reactant forms a process in random medium.

We describe asymptotically the genealogy of catalytic branching populations coded as the induced forest of ℝ-trees using the many individuals—rapid branching continuum limit. The limiting continuum genealogical forests are then studied in detail from both the quenched and annealed points of view. The result is obtained by constructing a contour process and analyzing the appropriately rescaled version and its limit. The genealogy of the limiting forest is described by a point process. We compare geometric properties and statistics of the reactant limit forest with those of the “classical” forest.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 3 (2009), 1232-1272.

Dates
First available in Project Euclid: 15 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1245071025

Digital Object Identifier
doi:10.1214/08-AAP574

Mathematical Reviews number (MathSciNet)
MR2537365

Zentralblatt MATH identifier
1178.60057

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments 60B11: Probability theory on linear topological spaces [See also 28C20] 92D25: Population dynamics (general)

Keywords
Catalytic branching random trees contour process genealogical point processes R-trees Gromov–Hausdorff topology random evolution

Citation

Greven, Andreas; Popovic, Lea; Winter, Anita. Genealogy of catalytic branching models. Ann. Appl. Probab. 19 (2009), no. 3, 1232--1272. doi:10.1214/08-AAP574. https://projecteuclid.org/euclid.aoap/1245071025


Export citation

References

  • [1] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [2] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
  • [3] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [4] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.
  • [5] Athreya, S. and Winter, A. (2005). Spatial coupling of neutral measure-valued population models. Stochastic Process. Appl. 115 891–906.
  • [6] Chiswell, I. (2001). Introduction to Λ-Trees. World Scientific, River Edge, NJ.
  • [7] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
  • [8] Dress, A. W. M. (1984). Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces. Adv. in Math. 53 321–402.
  • [9] Dress, A. W. M., Moulton, V. and Terhalle, W. F. (1996). T-theorie. Europ. J. Combinatorics 17 161–175.
  • [10] Dress, A. W. M. and Terhalle, W. F. (1996). The real tree. Adv. in Math. 120 283–301.
  • [11] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi–147.
  • [12] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [13] Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81–126.
  • [14] Evans, S. N. and Winter, A. (2006). Subtree prune and re-graft: A reversible real-tree valued Markov chain. Ann. Probab. To appear. 34 918–961.
  • [15] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I, 3rd ed. Wiley, New York.
  • [16] Greven, A., Klenke, A. and Wakolbinger, A. (1999). The longtime behavior of branching random walk in a catalytic medium. Electron. J. Probab. 4 80.
  • [17] Kurtz, T. G. (1992). Averaging for martingale problems and stochastic approximation. In Applied Stochastic Analysis (New Brunswick, NJ, 1991). Lecture Notes in Control and Information Sciences 177 186–209. Springer, Berlin.
  • [18] Lamperti, J. (1967). The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 271–288.
  • [19] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
  • [20] Penssel, C. (2003). Interacting Feller diffusions in catalytic media. Ph.D. thesis, Institute of Math., Erlangen, Germany.
  • [21] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Springer, Berlin.
  • [22] Popovic, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Probab. 14 2120–2148.
  • [23] Protter, P. E. (1977). On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic integral equations. Ann. Probab. 5 243–261.
  • [24] Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales: Itô Calculus. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. 2. Wiley, New York.
  • [25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [26] Stroock, D. W. and Varadhan, S. R. S. (1969). Diffusion processes with continuous coefficients. I. Comm. Pure Appl. Math. 22 345–400.
  • [27] Terhalle, W. F. (1997). R-trees and symmetric differences of sets. European J. Combin. 18 825–833.