具有扩散和竞争的连续粒子系统的随机演化：微观和介观描述
Stochastic evolution of a continuum particle system with dispersal and competition: micro- and mesoscopic description
摘要：
在微观和介观水平上描述了$\mathbb{R}^d$点粒子系统的马尔可夫演化。这些粒子在遥远的点上复制自己（扩散），然后在彼此的影响下独立地死亡（竞争）。微观描述是基于一个无限链的关联函数方程，类似于BBGKY层次的连续粒子系统的哈密顿动力学。介观的描述是基于一个Vlasov型动力学方程的颗粒的密度从上述链通过缩放程序。微观理论的主要结论是竞争可以阻止系统的聚集，这使得用密度描述系统是合理的。还讨论了长时间极限下动力学方程解的均匀化问题。
英文摘要：
A Markov evolution of a system of point particles in $\mathbb{R}^d$ is described at micro-and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and under the influence of each other (competition). The microscopic description is based on an infinite chain of equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a Vlasov-type kinetic equation for the particle's density obtained from the mentioned chain via a scaling procedure. The main conclusion of the microscopic theory is that the competition can prevent the system from clustering, which makes its description in terms of densities reasonable. A possible homogenization of the solutions to the kinetic equation in the long-time limit is also discussed.
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