Abstract
The nested Kingman coalescent describes the dynamics of particles (called genes) contained in larger components (called species), where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species. We prove that starting from rn species, the empirical distribution of species masses (numbers of genes/n) at time t / n converges as \(n\rightarrow \infty \) to a solution of the deterministic coagulation-transport equation
where \(\psi (x) = cx^2\), \(\star \) denotes convolution and \(a(t)= 1/(t+\delta )\) with \(\delta =2/r\). The most interesting case when \(\delta =0\) corresponds to an infinite initial number of species. This equation describes the evolution of the distribution of species of mass x, where pairs of species can coalesce and each species’ mass evolves like \(\dot{x} = -\psi (x)\). We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when \(\delta =0\). The first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous-state branching processes. The second one is the law of the solution to the following McKean–Vlasov equation
where J is an inhomogeneous Poisson process with rate \(1/(t+\delta )\) and \((v_t; t\ge 0)\) is a sequence of independent random variables such that \({{\mathcal {L}}}(v_t) = {{\mathcal {L}}}(x_t)\). We show that there is a unique solution to this equation and we construct this solution with the help of a marked Brownian coalescent point process. When \(\psi (x)=x^\gamma \), we show the existence of a self-similar solution for the PDE which relates when \(\gamma =2\) to the speed of coming down from infinity of the nested Kingman coalescent.
This is a preview of subscription content, access via your institution.



References
Abulwafa, E., Abdou, M., Mahmoud, A.: The solution of nonlinear coagulation problem with mass loss. Chaos Solitons Fractals 29(2), 313–330 (2006)
Aldous, D.: Stopping times and tightness. Ann. Probab. 6(2), 335–340 (1978)
Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5(1), 3–48 (1999)
Berestycki, J., Berestycki, N., Limic, V.: The \(\Lambda \)-coalescent speed of coming down from infinity. Ann. Probab. 38(1), 207–233 (2010)
Berestycki, J., Berestycki, N., Schweinsberg, J.: Beta-coalescents and continuous stable random trees. Ann. Probab. 35, 1835–1887 (2007)
Berestycki, N.: Recent progress in coalescent theory. Ensaios Mat. 16(1), 1–193 (2009)
Bertoin, J.: Random Fragmentation and Coagulation Processes, vol. 102. Cambridge University Press, Cambridge (2006)
Bertoin, J., Le Gall, J.-F.: Stochastic flows associated to coalescent processes. Ill. J. Math. III Limit Theorems 50(1–4), 147–181 (2006)
Blancas, A., Duchamps, J.-J., Lambert, A., Siri-Jégousse, A.: Trees within trees: simple nested coalescents. Electron. J. Probab. 23(94), 1–27 (2018)
Blancas, A., Rogers, T., Schweinsberg, J., Siri-Jégousse, A.: The nested Kingman coalescent: speed of coming down from infinity. Ann. Appl. Probab. 29(3), 1808–1836 (2019)
Caballero, M.E., Lambert, A., Uribe Bravo, G.: Proof (s) of the Lamperti representation of continuous-state branching processes. Probab. Surv. 6, 62–89 (2009)
Derrida, B., Retaux, M.: The depinning transition in presence of disorder: a toy model. J. Stat. Phys. 156(2), 268–290 (2014)
Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi + 147 (2002)
Feller, W.: Two singular diffusion problems. Ann. Math. 54(1), 173–182 (1951)
Fournier, N., Méléard, S.: A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14, 1880–1919 (2004)
Grey, D.: Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11(4), 669–677 (1974)
Hu, Y., Mallein, B., Pain, M.: An exactly solvable continuous-time Derrida–Retaux model (2018). arXiv preprint arXiv:1811.08749
Joffe, A., Métivier, M.: Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Probab. 18(1), 20–65 (1986)
Kingman, J.F.C.: The coalescent. Stoch. Process. Their Appl. 13(3), 235–248 (1982)
Lambert, A.: Population dynamics and random genealogies. Stoch. Models 24(Suppl. 1), 45–163 (2008)
Lambert, A., Schertzer, E.: Recovering the Brownian coalescent point process from the Kingman coalescent by conditional sampling. Bernoulli 25, 148–173 (2019)
Lamperti, J.: Continuous state branching processes. Bull. Am. Math. Soc. 73(3), 382–386 (1967)
Norris, J.R.: Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9(1), 78–109 (1999)
Popovic, L.: Asymptotic genealogy of a critical branching process. Ann. Appl. Probab. 14(4), 2120–2148 (2004)
Roelly-Coppoletta, S.: A criterion of convergence of measure-valued processes: application to measure branching processes. Stoch. Int. J. Probab. Stoch. Process. 17(1–2), 43–65 (1986)
Singh, P., Rodgers, G.: Coagulation processes with mass loss. J. Phys. A Math. Gen. 29(2), 437 (1996)
Sznitman, A.-S.: Topics in propagation of chaos. In: Ecole d’été de probabilités de Saint-Flour XIX—1989, pp. 165–251. Springer (1991)
Tran, V.C.: Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM Probab. Stat. 12, 345–386 (2008)
Tran, V.C.: Une ballade en forêts aléatoires. Technical report (2014)
Wattis, J.A., McCartney, D.G., Gudmundsson, T.: Coagulation equations with mass loss. J. Eng. Math. 49(2), 113–131 (2004)
Wennberg, B.: An example of non-uniqueness for solutions to the homogeneous Boltzmann equation. J. Stat. Phys. 95(1–2), 469–477 (1999)
Acknowledgements
The authors thank Center for Interdisciplinary Research in Biology (CIRB, Collège de France) for funding. We would like to thank the referee for her/his thorough review. We highly appreciated her/his comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A
Appendix A
Here, we complete the proof of Theorem 2.4 (ii) by showing that \(\mu _T\)defined as\(F(\mathbf{T}, (W_i); 1\le i \le N_T (\mathbf T))\) indeed is a solution to (1.3).
Let f be a test-function as defined before Definition 1.8, i.e., \(f\in {{\mathcal {C}}}^1({{\mathbb {R}}}^+)\) such that f and \(f' \psi \) are bounded. Hereafter, we continue to denote by \(\mathbb {P}_T\) the joint law of the pure-birth tree \(\mathbf{T}\) started with one particle at time 0, birth rate \(a(T-t)\), stopped at time T, and of the iid random variables \((W_i; 1\le i \le N_T (\mathbf T))\) with law \(\nu \). We will abbreviate \(F(\mathbf{T}, (W_i); 1\le i \le N_T (\mathbf T))\) into \(F(\mathbf{T})\). In particular, denoting \(\mu _T\) as the law of \(F(\mathbf{T}, (W_i); 1\le i \le N_T (\mathbf T))\) under \(\mathbb {P}_T\), we have
so that
where \(\mathbf{T}'\) is an independent copy of \(\mathbf{T}\) and \(\mathbf{T}+\mathbf{T}'\) denotes the tree splitting at time 0 into the two subtrees \(\mathbf{T}\) and \(\mathbf{T}'\). First note that by construction, \(F(\mathbb {t}+\mathbb {t}')= F(\mathbb {t}) + F(\mathbb {t}')\). Second, if we denote by \(\mathbb {t}+\varepsilon \) the tree obtained from \(\mathbb {t}\) by merely adding a length \(\varepsilon \) to its root edge, then by the Markov property of the entrance measure of the CSBP at 0,
Now as specified at the end of Sect. 2.3, for each fixed \(\lambda \), \(N\left( 1-\exp \left( -\lambda Z_t\right) \right) \) is solution to \(\dot{x} =-\psi (x)\) with initial condition \(x(0)=\lambda \). As a consequence,
Combining the last two results, we obtain
Next, since \(\psi f'\) is bounded, by dominated convergence, we get
As a consequence,
So \(t\mapsto \mu _t(f)\) is right-differentiable with continuous right-derivative equal to
Also note that
so that \(F(\varnothing )= w_1\) and \(\mu _0(f)= \mathbb {E}_0(f(F(\mathbf{T})))=\mathbb {E}_0(f(W))=\nu (f)\). This shows that \(\mu _0=\nu \) so that \((\mu _t(f);t\ge 0)\) satisfies (1.5).
Rights and permissions
About this article
Cite this article
Lambert, A., Schertzer, E. Coagulation-transport equations and the nested coalescents. Probab. Theory Relat. Fields 176, 77–147 (2020). https://doi.org/10.1007/s00440-019-00914-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-019-00914-4
Keywords
- Kingman coalescent
- Smoluchowski equation
- McKean–Vlasov equation
- Degenerate PDE
- PDE probabilistic solution
- Hydrodynamic limit
- Entrance boundary
- Empirical measure
- Coalescent point process
- Continuous-state branching process
- Phylogenetics
Mathematics Subject Classification
- Primary 60K35
- Secondary 35Q91
- 35R09
- 60G09
- 60B10
- 60G55
- 60G57
- 60J25
- 60J75
- 60J80
- 62G30
- 92D15