Gaussian fluctuations of the moderately interacting Shigesada-Kawasaki-Teramoto particle system around the mollified mean-field limit

Abstract

We study, at the logarithmic mollification scale $η^{-2(d+1+α)}\leδ\log N$ of Chen, Daus, Holzinger and J{ü}ngel, the fluctuation field $V^N=\sqrt N(μ^N-u_η)$ of the moderately interacting multi-species particle system whose mean-field limit is the Shigesada- Kawasaki-Teramoto (SKT) cross-diffusion system. The field is centred at the solution $u_η$ of the mollified equation, which depends on $N$ through $η$. On a scale of weighted Bessel-potential spaces of negative order we prove that the laws of $(V^N,M^N)$ are tight, where $M^N$ is the martingale part. Every limit point of $M^N$ is a continuous Gaussian martingale with an explicit deterministic bracket. Every limit point of $V^N$ is a weak solution, and the mild solution, of a linear Ornstein-Uhlenbeck SPDE whose drift is the linearisation of the SKT operator at the limit $u$. The initial field and the martingale are independent, and weak solutions are unique given both, so the whole sequence converges in law, jointly with the martingale part, and the limit is unique in law among solutions of the martingale problem. The noise is $d M_i=div(\sqrt{2u_iΘ_i}dW_i)$, with $Θ_i=σ_i+\sum_jf(a_{ij}u_j)$ and $W$ a cylindrical Wiener process on $L^2(\mathbb{R}^d;\mathbb{R}^{nd})$. The hypotheses are those of Chen, Daus, Holzinger and J{ü}ngel, including the smallness and the nonnegativity on $\mathbb{R}$ of the nonlinearity $f$. We add radial kernels $B_{ij}$, the independence of the initial data and the Brownian motions across particles and species, a smaller $δ$, a regularity budget on $u_0$, a strengthened smallness condition on $f$ that makes the backward system diagonally dominant in weighted spaces, and an integer Sobolev index $s$.