Supercritical McKean-Vlasov SDE driven by cylindrical $α$-stable process
Zimo Hao, Chongyang Ren, Mingyan Wu
Abstract
In this paper, we study the following supercritical McKean-Vlasov SDE, driven by a symmetric non-degenerate cylindrical $α$-stable process in $\mathbb{R}^d$ with $α\in (0,1)$: $$ \mathord{\rm d} X_t = (K * μ_{t})(X_t)\mathord{\rm d}t + \mathord{\rm d} L_t^{(α)}, \quad X_0 = x \in \mathbb{R}^d, $$ where $K: \mathbb{R}^d \to \mathbb{R}^d$ is a $β$-order Hölder continuous function, and $μ_t$ represents the time marginal distribution of the solution $X$. We establish both strong and weak well-posedness under the conditions $β\in (1 - α/2, 1)$ and $β\in (1 - α, 1)$, 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.