凝并-输运方程和嵌套聚结
Coagulation-transport equations and the nested coalescents
摘要：
嵌套的Kingman聚结描述了包含在较大组分（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聚结，并且基因对以恒定速率聚结，只要它们位于同一物种内。我们证明，从$rn$物种，物种质量的经验分布（基因$/n$的数量）在时间$t/n$收敛为$n\to\infty$的确定性的凝固-运输方程$$ \partial_t d \ = \ \partial_x ( ψd ) \ + \ a(t)\left(d\star d - d \right), $$的解决方案，其中$ψ(x) = cx^2$，$\star$表示卷积和$a(t)= 1/(t+δ)$与$δ=2/r$。最有趣的情况下，当$δ=0$对应于一个无限的初始数量的物种。该方程描述了质量为$x$的物质分布的演化，其中成对的物质可以合并，并且每个物质的质量像$\dot x = -ψ(x)$一样演化。我们提供了两个自然的概率解决方案，后者IPDE和地址详细的情况下，$δ=0$。第一个解决方案是表示在一个分支粒子系统中的粒子携带的质量表现为独立的连续状态分支过程。第二个是McKean-Vlasov方程$$ dx_t \ = \ - ψ(x_t) \,dt \ + \ v_t\,ΔJ_t $$的解的规律，其中$J$是一个具有速率$1/(t+δ)$的非齐次Poisson过程，$(v_t; t\geq0)$是一个独立的rvs序列，使得$ {\mathcal L}(v_t) = {\mathcal L}(x_t)$ .我们证明了这个方程存在唯一解，并利用一个显着的布朗结合点过程构造了这个解。当$ψ(x)=x^γ$时，我们证明了PDE的自相似解的存在性，当$γ=2$与嵌套Kingman聚结从无穷远下降的速度有关。
英文摘要：
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 $n\to\infty$ to a solution of the deterministic coagulation-transport equation $$ \partial_t d \ = \ \partial_x ( ψd ) \ + \ a(t)\left(d\star d - d \right), $$ where $ψ(x) = cx^2$, $\star$ denotes convolution and $a(t)= 1/(t+δ)$ with $δ=2/r$. The most interesting case when $δ=0$ 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 $\dot x = -ψ(x)$. We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when $δ=0$. 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 $$ dx_t \ = \ - ψ(x_t) \,dt \ + \ v_t\,ΔJ_t $$ where $J$ is an inhomogeneous Poisson process with rate $1/(t+δ)$ and $(v_t; t\geq0)$ is a sequence of independent rvs 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 $ψ(x)=x^γ$, we show the existence of a self-similar solution for the PDE which relates when $γ=2$ to the speed of coming down from infinity of the nested Kingman coalescent.
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