论文标题
具有延迟和随机延迟微分方程的交互粒子系统
Interacting particles systems with delay and random delay differential equations
论文作者
论文摘要
在这项工作中，我们研究了具有延迟动力学的活动粒子的动力学模型，当颗粒数量进入无穷大时，其极限。事实证明，该限制与具有随机初始条件的延迟微分方程有关。我们分析了两个不同的动力学，一种基于每个粒子各个轨迹的全部知识，而另一个仅基于粒子云的轨迹，失去了单个轨迹的轨迹。请注意，在第一个动态中，粒子的状态是其路径，而在第二种情况下，它只是$ \ r^d $中的一点。在这两种情况下，我们分析了相应的平均场动力学，以获取粒子状态分布时间演变的方程。方程式的适合性通过定点参数证明。我们以一些可能的未来研究方向和建模应用结束了本文。
In this work we study a kinetic model of active particles with delayed dynamics, and its limit when the number of particles goes to infinity. This limit turns out to be related to delayed differential equations with random initial conditions. We analyze two different dynamics, one based on the full knowledge of the individual trajectories of each particle, and another one based only on the trace of the particle cloud, loosing track of the individual trajectories. Notice that in the first dynamic the state of a particles is its path, whereas it is simply a point in $\R^d$ in the second case. We analyse in both cases the corresponding mean-field dynamic obtaining an equation for the time evolution of the distribution of the particles states. Well-posedness of the equation is proved by a fixed-point argument. We conclude the paper with some possible future research directions and modelling applications.
