复制子-突变子方程的概率分析
Probabilistic Analysis of Replicator-Mutator Equations
摘要：
本文在多维适应度空间上引入了一类一般的复制-变异方程。利用Fockker-Planck-Kolmogorov（FPK）方程理论和鞅抽取方法，建立了该方程弱解的一种新的概率表示.给出了现有文献中考虑的不同适应度函数的封闭形式概率解的例子。我们还构建了一个粒子系统，并证明了一个一般的收敛结果，任何解决方案的FPK方程与扩展的复制突变方程相对于一个Wasserstein的距离适应我们的概率框架。
英文摘要：
This paper introduces a general class of Replicator-Mutator equations on a multi-dimensional fitness space. We establish a novel probabilistic representation of weak solutions of the equation by using the theory of Fockker-Planck-Kolmogorov (FPK) equations and a martingale extraction approach. The examples with closed-form probabilistic solutions for different fitness functions considered in the existing literature are provided. We also construct a particle system and prove a general convergence result to any solution to the FPK equation associated with the extended Replicator-Mutator equation with respect to a Wasserstein-like distance adapted to our probabilistic framework.
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