Mean-field limit of collective dynamics with time-varying weights
Mean-field limit of collective dynamics with time-varying weights
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随时间变化权重的集体动力学的平均场极限
DOI：
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发表时间：
2021
期刊：
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通讯作者：
N. Duteil
中科院分区：
文献类型：
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作者：
N. Duteil
在这篇文章中，我们导出了一个具有时变权的集体动力学模型的平均场极限，对于保持系统总质量和主体的不可区分性的权重动力学。极限方程是一个有源的输运方程，其中(非局部)输运项对应于位置动力学，而(非局部)源项来自于智能体之间的权重重新分配。我们证明了微观模型和宏观模型解的存在唯一性，并引入了一种新的考虑权重的经验度量。在Wasserstein拓扑和有界Lipschitz拓扑中，通过证明宏观解关于初始数据的连续性，我们得到了微观模型向宏观模型的收敛。
In this paper, we derive the mean-field limit of a collective dynamics model with time-varying weights, for weight dynamics that preserve the total mass of the system as well as indistinguishability of the agents. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution for both microscopic and macroscopic models and introduce a new empirical measure taking into account the weights. We obtain the convergence of the microscopic model to the macroscopic one by showing continuity of the macroscopic solution with respect to the initial data, in the Wasserstein and Bounded Lipschitz topologies.