We study a class of McKean--Vlasov Stochastic Differential Equations (MV-SDEs) with drifts and diffusions having super-linear growth in measure and space -- the maps have general polynomial form but also satisfy a certain monotonicity condition. The combination of the drift's super-linear growth in measure (by way of a convolution) and the super-linear growth in space and measure of the diffusion coefficient require novel technical elements in order to obtain the main results. We establish wellposedness, propagation of chaos (PoC), and under further assumptions on the model parameters we show an exponential ergodicity property alongside the existence of an invariant distribution. No differentiability or non-degeneracy conditions are required. Further, we present a particle system based Euler-type split-step scheme (SSM) for the simulation of this type of MV-SDEs. The scheme attains, in stepsize, the strong error rate $1/2$ in the non-path-space root-mean-square error metric and we demonstrate the property of mean-square contraction. Our results are illustrated by numerical examples including: estimation of PoC rates across dimensions, preservation of periodic phase-space, and the observation that taming appears to be not a suitable method unless strong dissipativity is present.

翻译：本文研究一类漂移项和扩散项均具有关于测度和空间超线性增长特征的McKean-Vlasov随机微分方程（MV-SDEs）——该类映射满足一般多项式形式的同时亦需满足特定单调性条件。漂移项关于测度的超线性增长（通过卷积实现）与扩散系数在空间和测度上的超线性增长的耦合效应，要求引入新颖的技术要素以获得主要结论。我们建立了适定性、混沌传播性质，并在模型参数附加假设下证明了指数遍历性及不变分布的存在性，且无需可微性或非退化性条件。进一步，提出了基于粒子系统的欧拉型分裂步格式（SSM）用于此类MV-SDEs的模拟。该格式在非路径空间均方根误差度量下达到步长$1/2$的强误差阶，并展现出均方压缩性质。数值算例验证了理论结果，包括：跨维度混沌传播速率的估计、相空间周期性的保持，以及观察到若无强耗散性存在，驯化算法并非适宜方法的特性。