Supercritical McKean-Vlasov SDE driven by cylindrical $\alpha$-stable process

Abstract

In this paper, we study the following supercritical McKean-Vlasov SDE, driven by a symmetric non-degenerate cylindrical $\alpha$-stable process in ${\mathbb{R}}^d$ with $\alpha \in (0,1)$: $$\mathrm{d}{X}_t=(K\ast {\mu }_t)(X_t)\mathrm{d}t+\mathrm{d}{L}_t^{(\alpha )},X_0=x\in {\mathbb{R}}^d,$$ where $K:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is a $\beta$-order Hölder continuous function, and ${\mu }_t$ represents the time marginal distribution of the solution $X$. We establish both strong and weak well-posedness under the conditions $\beta \in (1-\alpha /2,1)$ and $\beta \in (1-\alpha ,1)$, respectively. Additionally, we demonstrate strong propagation of chaos for the associated interacting particle system, as well as the convergence of the corresponding Euler approximations. In particular, we prove a commutation property between the particle approximation and the Euler approximation.

36 pages, 1 figure