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 requires 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与空间上均具有超线性增长特性的McKean--Vlasov随机微分方程（MV-SDE）——该映射具有一般多项式形式，同时满足特定的单调性条件。漂移项通过卷积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，需要引入新的技术要素以获得主要结果。我们证明了方程的适定性、混沌传播性（PoC），并在进一步假设模型参数的前提下，展示了指数遍历性及不变分布的存在性。该结果不要求可微性或非退化性条件。此外，我们提出一种基于粒子系统的欧拉型分裂步格式（SSM）用于此类MV-SDE的数值模拟。该格式在步长意义下，于非路径空间均方根误差度量中达到$1/2$的强误差阶，并证明了均方压缩性质。数值算例展示了以下结果：跨维度混沌传播速率的估计、周期相空间的保持特性，以及除非存在强耗散性否则驯服法并不适用的现象。