Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming
Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming
复制标题
随机平均场极限：非 Lipschitz 力
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Francois Bolley Jos
中科院分区：
文献类型：
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作者：
Francois Bolley Jos
我们认为一般的随机系统的相互作用粒子的噪声是相关的动物的集体行为的模型，并严格证明，在平均场的限制，该系统是接近的动力学PDE的解决方案。我们的目标是包括在文献中广泛研究的模型，如CuckerSmale模型，为个人行为添加噪音。与经典的全局Lipschitz势的情形相比，其困难在于在几种模型中粒子间的相互作用势仅为局部Lipschitz，局部Lipschitz常数随着所考虑区域的大小而无穷大.考虑到这一点，我们提出了一个扩展的经典理论的全局Lipschitz相互作用，它只适用于局部Lipschitz的。
We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solutio n of a kinetic PDE. Our aim is to include models widely studied in the literature such as the CuckerSmale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is t hat in several models the interaction potential between particles is only locally Li pschitz, the local Lipschitz constant growing to infinity with the size of the region consi dered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.