稳定过程驱动的McKean-Vlasov随机微分方程的混沌定量弱传播
Quantitative weak propagation of chaos for stable-driven McKean-Vlasov SDEs
AI总结：
本文研究旋转不变α-稳定过程驱动的McKean-Vlasov随机微分方程，利用对应半群的正则性与倒向Kolmogorov方程描述的动力学，在半群层面证明了平均场相互作用粒子系统的新混沌传播估计。
摘要：
我们研究一类由$\u211d^d$上旋转不变$\u03b1$-稳定过程驱动的一般McKean-Vlasov随机微分方程，其中$\u03b1\u2208(1,2)$。假设扩散系数为单位矩阵，漂移项有界且在空间和测度变量的某种精确意义下满足Hölder连续。本文的主要目标是在半群层面，为对应的平均场相互作用粒子系统证明新的混沌传播估计。我们的研究依赖于McKean-Vlasov随机微分方程对应半群的正则化性质与动力学，该半群作用于定义在$\u2119_\u03b2(\u211d^d)$上的函数，$\u2119_\u03b2(\u211d^d)$是$\u211d^d$上具有有限$\u03b2$阶矩的概率测度空间，其中$\u03b2\u2208(1,\u03b1)$。更具体地说，半群的动力学由定义在带状区域$[0,T]\times\u2119_\u03b2(\u211d^d)$上的倒向Kolmogorov偏微分方程描述。
英文摘要：
We consider a general McKean-Vlasov stochastic differential equation driven by a rotationally invariant $α$-stable process on $\mathbb{R}^d$ with $α\in (1,2)$. We assume that the diffusion coefficient is the identity matrix and that the drift is bounded and H{ö}lder continuous in some precise sense with respect to both space and measure variables. The main goal of this work is to prove new propagation of chaos estimates, at the level of semigroup, for the associated mean-field interacting particle system. Our study relies on the regularizing properties and the dynamics of the semigroup associated with the McKean-Vlasov stochastic differential equation, which acts on functions defined on $\mathcal{P}_β(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$ having a finite moment of order $β\in (1,α)$. More precisely, the dynamics of the semigroup is described by a backward Kolmogorov partial differential equation defined on the strip $[0,T] \times \mathcal{P}_β(\mathbb{R}^d)$.
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