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Coagulation-transport equations and the nested coalescents

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

$$\begin{aligned} \partial _t d \ = \ \partial _x ( \psi d ) \ + \ a(t)\left( d\,\star \,d - d \right) , \end{aligned}$$

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

$$\begin{aligned} dx_t \ = \ - \psi (x_t) \,dt \ + \ v_t\,\Delta J_t \end{aligned}$$

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.

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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.

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Correspondence to Amaury Lambert.

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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

$$\begin{aligned} \mu _T(f):=\int _{{{\mathbb {R}}}^+} f(x)\, \mu _T(dx) = \mathbb {E}_T( f\circ F(\mathbf{T})), \end{aligned}$$

so that

$$\begin{aligned} \mu _{T+\varepsilon }(f) = \mathbb {E}_{T+\varepsilon }( f\circ F(\mathbf{T}), N_\varepsilon =1) + a(T)\varepsilon \, \mathbb {E}_T^{\otimes 2}( f\circ F(\mathbf{T}+\mathbf{T}'))+o(\varepsilon ), \end{aligned}$$

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,

$$\begin{aligned} F(\mathbb {t}+\varepsilon )= & {} M^{\mathbb {t}+\varepsilon }\left( 1-\exp \left( -\sum _{i=1}^{N_T(\mathbb {t})}w_iZ_{T+\varepsilon }^i\right) \right) \\= & {} \int _{(0,\infty )}N(Z_\varepsilon \in dx )\, Q_x^\mathbb {t}\left( 1-\exp \left( -\sum _{i=1}^{N_T(\mathbb {t})}w_iZ_T^i\right) \right) \\= & {} \int _{(0,\infty )}N(Z_\varepsilon \in dx ) \left( 1-\exp \left( -xM^\mathbb {t}\left( 1-\exp \left( -\sum _{i=1}^{N_T(\mathbb {t})}w_iZ_T^i\right) \right) \right) \right) \\= & {} N\left( 1-\exp \left( -Z_\varepsilon F(\mathbb {t})\right) \right) . \end{aligned}$$

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,

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\varepsilon ^{-1}\left( F(\mathbb {t}+\varepsilon )- F(\mathbb {t})\right) =\lim _{\varepsilon \downarrow 0}\varepsilon ^{-1}\left( N\left( 1-\exp \left( -Z_\varepsilon F(\mathbb {t})\right) \right) -F(\mathbb {t})\right) =-\psi (F(\mathbb {t})). \end{aligned}$$

Combining the last two results, we obtain

$$\begin{aligned} \mu _{T+\varepsilon }(f)= & {} \mathbb {E}_{T+\varepsilon }( f\circ F(\mathbf{T}), N_\varepsilon =1) + a(T)\varepsilon \, \mathbb {E}_T^{\otimes 2}( f\circ F(\mathbf{T}+\mathbf{T}'))+o(\varepsilon )\\= & {} (1-a(T)\varepsilon )\,\mathbb {E}_{T}( f\circ F(\mathbf{T}+\varepsilon ))\\&+ a(T)\varepsilon \, \mathbb {E}_T^{\otimes 2}( f\circ (F(\mathbf{T})+F(\mathbf{T}')))+o(\varepsilon )\\= & {} \mu _{T}(f) + (1-a(T)\varepsilon )\, \mathbb {E}_{T}( f\circ F(\mathbf{T}+\varepsilon )- f\circ F(\mathbf{T})) \\&+a(T)\varepsilon \, (\mu _T^{\star 2}(f)- \mu _T(f))+o(\varepsilon ). \end{aligned}$$

Next, since \(\psi f'\) is bounded, by dominated convergence, we get

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\varepsilon ^{-1}\mathbb {E}_{T}( f\circ F(\mathbf{T}+\varepsilon )- f\circ F(\mathbf{T})) = -\mathbb {E}_{T}\big (\psi (F(\mathbf{T}))\, f'(F(\mathbf{T}))\big )=-\mu _T(\psi f'). \end{aligned}$$

As a consequence,

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\varepsilon ^{-1}(\mu _{T+\varepsilon }(f)- \mu _{T}(f)) = -\mu _T(\psi f')+a(T)\, (\mu _T^{\star 2}(f)- \mu _T(f)). \end{aligned}$$

So \(t\mapsto \mu _t(f)\) is right-differentiable with continuous right-derivative equal to

$$\begin{aligned} \partial _t \mu _t(f) = -\mu _t(\psi f')+a(t)\, (\mu _t^{\star 2}(f)- \mu _t(f))\qquad t\ge 0. \end{aligned}$$

Also note that

$$\begin{aligned} F(\varnothing +\varepsilon ) = M^{\varnothing +\varepsilon }\left( 1-\exp \left( -w_1Z_\varepsilon \right) \right) =N \left( 1-\exp \left( -w_1Z_\varepsilon \right) \right) , \end{aligned}$$

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).

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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

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  • 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