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 $$n\rightarrow \infty $$ n → ∞ to a solution of the deterministic coagulation-transport equation $$\begin{aligned} \partial _t d \ = \ \partial _x ( \psi d ) \ + \ a(t)\left( d\,\star \,d - d \right) , \end{aligned}$$ ∂ t d = ∂ x ( ψ d ) + a ( t ) d ⋆ d - d , where $$\psi (x) = cx^2$$ ψ ( x ) = c x 2 , $$\star $$ ⋆ denotes convolution and $$a(t)= 1/(t+\delta )$$ a ( t ) = 1 / ( t + δ ) with $$\delta =2/r$$ δ = 2 / r . The most interesting case when $$\delta =0$$ δ = 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 $$\dot{x} = -\psi (x)$$ x ˙ = - ψ ( x ) . We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when $$\delta =0$$ δ = 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 $$\begin{aligned} dx_t \ = \ - \psi (x_t) \,dt \ + \ v_t\,\Delta J_t \end{aligned}$$ d x t = - ψ ( x t ) d t + v t Δ J t where J is an inhomogeneous Poisson process with rate $$1/(t+\delta )$$ 1 / ( t + δ ) and $$(v_t; t\ge 0)$$ ( v t ; t ≥ 0 ) is a sequence of independent random variables such that $${{\mathcal {L}}}(v_t) = {{\mathcal {L}}}(x_t)$$ L ( v t ) = 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 $$\psi (x)=x^\gamma $$ ψ ( x ) = x γ , we show the existence of a self-similar solution for the PDE which relates when $$\gamma =2$$ γ = 2 to the speed of coming down from infinity of the nested Kingman coalescent.

中文翻译：

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凝结输运方程和嵌套聚结

嵌套的 Kingman 聚结描述了包含在较大成分（称为物种）中的粒子（称为基因）的动力学，其中物种对以恒定速率合并，而基因对以恒定速率合并，前提是它们位于同一物种内。我们证明从 rn 个物种开始，物种质量的经验分布（基因数/ n ）在时间 t / n 收敛为 $$n\rightarrow \infty $$ n → ∞ 到确定性凝血传输方程的解$$\begin{aligned} \partial _t d \ = \ \partial _x ( \psi d ) \ + \ a(t)\left( d\,\star \,d - d \right) , \end{对齐}$$ ∂ td = ∂ x ( ψ d ) + a ( t ) d ⋆ d - d ，其中$$\psi (x) = cx^2$$ ψ ( x ) = cx 2 , $$\star $ $ ⋆ 表示卷积，$$a(t)= 1/(t+\delta )$$ a ( t ) = 1 / ( t + δ )，其中 $$\delta =2/r$$ δ = 2 / r 。当 $$\delta =0$$ δ = 0 时最有趣的情况对应于无限的初始物种数。这个方程描述了质量为 x 的物种分布的演化，其中物种对可以合并，每个物种的质量演化如 $$\dot{x} = -\psi (x)$$ x ˙ = - ψ ( x )。我们提供了后者 IPDE 的两个自然概率解，并详细解决了 $$\delta =0$$ δ = 0 的情况。第一个解决方案用分支粒子系统表示，其中粒子携带质量，表现为独立的连续状态分支过程。第二个是以下 McKean-Vlasov 方程 $$\begin{aligned} dx_t \ = \ - \psi (x_t) \,dt \ + \ v_t\ 的解法，\Delta J_t \end{aligned}$$ dxt = - ψ ( xt ) dt + vt Δ J t 其中 J 是非齐次泊松过程，速率为 $$1/(t+\delta )$$ 1 / ( t + δ ) 和$$(v_t; t\ge 0)$$ ( vt ; t ≥ 0 ) 是一个独立的随机变量序列，使得 $${{\mathcal {L}}}(v_t) = {{\mathcal {L} }}(x_t)$$ L ( vt ) = L ( xt ) 。我们证明了这个方程有一个唯一的解，我们在一个标记的布朗聚结点过程的帮助下构造了这个解。当 $$\psi (x)=x^\gamma $$ ψ ( x ) = x γ 时，我们证明了当 $$\gamma =2$$ γ = 2 时 PDE 的自相似解的存在从嵌套金曼聚结的无穷远下降的速度。t ≥ 0 ) 是一系列独立的随机变量，使得 $${{\mathcal {L}}}(v_t) = {{\mathcal {L}}}(x_t)$$ L ( vt ) = L ( xt )。我们证明了这个方程有一个唯一的解，我们在一个标记的布朗聚结点过程的帮助下构造了这个解。当 $$\psi (x)=x^\gamma $$ ψ ( x ) = x γ 时，我们证明了当 $$\gamma =2$$ γ = 2 时 PDE 的自相似解的存在从嵌套金曼聚结的无穷远下降的速度。t ≥ 0 ) 是一系列独立的随机变量，使得 $${{\mathcal {L}}}(v_t) = {{\mathcal {L}}}(x_t)$$ L ( vt ) = L ( xt )。我们证明了这个方程有一个唯一的解，我们在一个标记的布朗聚结点过程的帮助下构造了这个解。当 $$\psi (x)=x^\gamma $$ ψ ( x ) = x γ 时，我们证明了当 $$\gamma =2$$ γ = 2 时 PDE 的自相似解的存在从嵌套金曼聚结的无穷远下降的速度。