Quantitative approximation of the Burgers and Keller-Segel equations by moderately interacting particles

In this work we obtain rates of convergence for two moderately interacting stochastic particle systems with singular kernels associated to the viscous Burgers and Keller-Segel equations. The main novelty of this work is to consider a non-locally integrable kernel. Namely for the viscous Burgers equation in $\mathbb{R}$, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some Bessel space with a rate of convergence of order $N^{-1/6}$, on any time interval. With the same rate, convergence also holds for the genuine empirical measure in Wasserstein distance, and at the level of the trajectories of the particles with the standard coupling to McKean-Vlasov particles. In the case of the Keller-Segel equation on a $d$-dimensional torus, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some $L^q$ space with a rate of order $N^{-\frac{1}{2(d+1)}}$. The result holds up to the maximal existence time of the PDE, for any value of the chemo-attractant sensitivity $\chi$.