摘要翻译：
我们研究了当扩散数趋于无穷大时，通过它们的秩相互作用的扩散系统的经验测度的极限行为。我们证明了极限动力学是由一个McKean-Vlasov发展方程给出的。此外，我们还证明了在许多情况下，极限动力学下累积分布函数的演化受含对流的广义多孔介质方程的控制。后者的唯一性理论被用来建立极限McKean-Vlasov方程解的唯一性和相应的相互作用扩散系统的大数定律。本文还解释了这些结果对金融市场中基于秩的资本分布模型的影响。
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英文标题：
《Large systems of diffusions interacting through their ranks》
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作者：
Mykhaylo Shkolnikov
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最新提交年份：
2010
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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英文摘要：
  We study the limiting behaviour of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that in a wide range of cases the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The uniqueness theory for the latter is used to establish the uniqueness of solutions of the limiting McKean-Vlasov equation and the law of large numbers for the corresponding systems of interacting diffusions. The implications of the results for rank-based models of capital distributions in financial markets are also explained.
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PDF链接：
https://arxiv.org/pdf/1008.4611