Supercritical McKean-Vlasov SDE driven by cylindrical α-stable process

2024/10/24 by Hao, Zimo, Ren, Chongyang, Wu, Mingyan
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2410.18611

Abstract

In this paper, we study the following supercritical McKean-Vlasov SDE, driven by a symmetric non-degenerate cylindrical α-stable process in ℝ^d with α∈ (0,1): \mathord\rm d X_t = (K * μ_t)(X_t)\mathord\rm dt + \mathord\rm d L_t^(α), X₀ = x ∈ ℝ^d, where K: ℝ^d → 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∈ (1 - α/2, 1) and β∈ (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.

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