Quantitative approximation to density dependent SDEs driven by $α$-stable processes
Ke Song, Zimo Hao, Mingkun Ye
TL;DR
The paper addresses quantitative approximation of density-dependent SDEs with $\alpha$-stable noise (McKean–Vlasov type) by moderately interacting $N$-particle systems. It combines Duhamel's formula, density estimates in Besov/Hölder spaces, and martingale inequalities to derive convergence rates for the empirical density and for single-particle marginals, with bounds that are robust to the noise type. Under bounded, Hölder drift and suitable initial data, the density error scales as $\|\rho_t-\rho_t^N\|_{L^m(\Omega;L^\infty)} \lesssim t^{-(\beta+d/q)/\alpha}N^{-\theta\beta}+N^{-1/2+\theta d+\varepsilon}$ (with improvements when $\rho_0\in C^\beta$), while weak and pathwise convergence of the first marginal follow with similar $N$-dependent rates. The methods extend propagation of chaos results to nonlocal, jump-driven dynamics and provide a rigorous bridge between the particle system and the nonlinear, nonlocal PDE $\partial_t\rho_t = \Delta^{\alpha/2}\rho_t - \mathrm{div}(b(t,x,\rho_t(x))\rho_t)$, highlighting a noise-type independent quantitative regime and yielding potential numerical and modeling benefits in physics, biology, and diffusion-based learning models.
Abstract
Based on a class of moderately interacting particle systems, we establish a quantitative approximation for density-dependent McKean-Vlasov SDEs and the corresponding nonlinear, nonlocal PDEs. The SDE is driven by both Brownian motion and pure-jump Lévy processes. By employing Duhamel's formula, density estimates, and appropriate martingale functional inequalities, we derive precise convergence rates for the empirical measure of particle systems toward the law of the McKean-Vlasov SDE solution. Additionally, we quantify both weak and pathwise convergence between the one-marginal particle and the solution to the McKean-Vlasov SDE. Notably, all convergence rates remain independent of the noise type.