Quantitative weak propagation of chaos for stable-driven McKean-Vlasov SDEs

We consider a general McKean-Vlasov stochastic differential equation driven by a rotationally invariant $\alpha$-stable process on $\mathbb{R}^d$ with $\alpha \in (1,2)$. We assume that the diffusion coefficient is the identity matrix and that the drift is bounded and H{\"o}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 $\mathcal{P}_{\beta}(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$ having a finite moment of order $\beta \in (1,\alpha)$. More precisely, the dynamics of the semigroup is described by a backward Kolmogorov partial differential equation defined on the strip $[0,T] \times \mathcal{P}_{\beta}(\mathbb{R}^d)$.