为解决大系统中粒子间的相互作用问题，Buckdahn等提出了平均场随机微分方程，成为近年来的研究热点。由于已有文献讨论的都是布朗运动驱动的连续轨道情形，而自然界很多系统除了受随机噪声影响，还可能受突发因素影响，因此，本项目拟提出带跳的(中立型)平均场泛函随机微分方程，并对其进行系统分析。内容包括：①提出Lévy过程驱动的(中立型)平均场泛函随机微分方程，并对解的存在唯一性、强(弱)稳定性等进行分析。②讨论Lévy过程驱动的(中立型)平均场泛函随机微分方程不变测度的存在唯一性和对应片段过程的指数遍历性，并探讨不变测度的大偏差原理。③建立Lévy过程驱动的(中立型)平均场泛函随机微分方程的Harnack和对数Harnack不等式。④研究节点动力学含Lévy噪声的延迟复杂网络模型稳定性和同步行为，给出其判别依据。本项目不仅能丰富平均场随机微分方程现有理论，而且还能为其在复杂网络中的应用建立桥梁。

Mean-field stochastic differential equations (MFSDE), a hot topic at present, was put forward by Buckdahn et al. in order to deal with the interactions among particles. However, all of the existing literature on MFSDE were based on Brownian motion, and, except for stochastic noises, many systems in nature may suffer sudden environmental shocks. Thus, in this project, we are going to have a systematic study on (neutral) mean-field stochastic functional differential equations with jumps. It mainly involves: ① (Neutral) Mean-field stochastic functional differential equations with jumps will be proposed, the existence and uniqueness, strong (weak) stability of the solutions will be analyzed. ② Invariant measures and stationary distributions of the (neutral) mean-field stochastic functional differential equations with jumps will be discussed, and large derivations of the invariant measures will be given. ③ Harnack inequality and log-Harnack inequality of the (neutral) mean-field stochastic functional differential equations with jumps will be derived. ④ Stability and synchronization conditions of delayed stochastic dynamical systems with jumps will be studied. This project will not only enrich the theories of MFSDE, but also establish a bridge between MFSDE and complex networks.