Uniform-in-time propagation of chaos for singular $L^k$ interaction kernels with rates

Abstract

Based on a semigroup approach, we prove quantitative uniform-in-time propagation of chaos for two systems of interacting particles with singular interaction kernels $K$ of type $L^k$, $k>d$, and a contractive Lipschitz confinement force field. First, we consider a system with moderate interaction, which means that the particles interact through the mollified kernel $K\ast V^N$, for some approximation of identity $V^N = N^{dα} V(N^α\cdot),\, α>0$. Assuming that $\|K\|_{L^k}$ is sufficiently small, we obtain that for any $\varepsilon>0$ the uniform-in-time rate is of order $N^{-\min\left(α,\frac12-\frac{dα}{2}-\varepsilon\right)}$, plus an exponentially small in time contribution of the initial condition. Second, for the classical, non-mollified particle system we establish uniform-in-time propagation of chaos with rate $N^{-\min\left(α,\frac12-\frac{dα}{2}-\varepsilon\right)} + \|K-K\ast V^N\|_{L^{k}}$, plus an exponentially small in time contribution of the initial condition. The method relies on the regularity properties of the solution of the associated nonlinear Fokker-Planck equation combined with estimates of the Duhamel formulation of the mollified empirical measure, together with partial Girsanov transforms for the non-mollified system of particles.