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

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 $V^N=N^{d\alpha }V(N^{\alpha }\cdot ),\alpha >0$. Assuming that $\mathrm{\parallel }K{\mathrm{\parallel }}_{L^k}$ is sufficiently small, we obtain that for any $\epsilon >0$ the uniform-in-time rate is of order $N^{-\min (\alpha ,\frac{1}{2}-\frac{d\alpha }{2}-\epsilon )}$, 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 $N^{-\min (\alpha ,\frac{1}{2}-\frac{d\alpha }{2}-\epsilon )}+\mathrm{\parallel }K-K\ast V^N{\mathrm{\parallel }}_{L^k}$, 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.