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在测度和空间上具有超线性增长（通过卷积实现）的McKean-Vlasov随机微分方程——映射具有一般多项式形式但满足特定单调性条件。漂移项在测度上的超线性增长与扩散系数在空间和测度上的超线性增长相结合，需要引入新颖的技术要素以获得主要结果。我们建立了该方程的适定性和混沌传播性质，并在模型参数进一步假设下，证明其指数遍历性及不变分布的存在性。该结论无需可微性或非退化性条件。此外，我们提出基于粒子系统的Euler型分步格式以模拟此类MV-SDEs。该格式在非路径空间均方根误差度量下达到步长$1/2$的强收敛阶，并具有均方收缩特性。数值实验验证了理论结果，包括：跨维度的混沌传播率估计、周期相空间保持，以及观察到除非存在强耗散性，驯服方法并不适用。