We consider a discrete model in which particles are characterized by two quantities [Formula: see text] and [Formula: see text]; both quantities evolve in time according to stochastic dynamics and the equation that governs the evolution of [Formula: see text] is also influenced by mean-field interaction between the particles. We allow for discontinuous coefficients and random initial condition and, under suitable assumptions, we prove that in the limit as the number of particles grows to infinity the dynamics of the system is described by the solution of a Fokker–Planck partial differential equation. We provide the existence and uniqueness of a solution to the latter and show that such solution arises as the limit in probability of the empirical measures of the system.

中文翻译：

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具有不连续系数及其连续极限的相互作用系统中的空间动力学

我们考虑一个离散模型，其中粒子由两个量[公式：见文本]和[公式：见文本]表征；这两个量都根据随机动力学随时间演变，并且控制[公式：见文本]演变的方程也受到粒子之间的平均场相互作用的影响。我们允许不连续的系数和随机的初始条件，并且在适当的假设下，我们证明了在极限情况下，随着粒子数增长到无穷大，系统的动力学可以通过 Fokker-Planck 偏微分方程的解来描述。我们为后者提供了解决方案的存在性和唯一性，并表明这种解决方案是作为系统经验度量的概率极限而出现的。