From Hamiltonian systems to compressible Euler equation driven by additive Hölder noise

In this paper, we derive stochastic compressible Euler Equation from a Hamiltonian microscopic dynamics and consider systems of interacting particles with Hölder noise and potential whose range is large in comparison with the typical distance between neighboring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of compressible Euler equations driven by additive Hölder path(noise), in the limit as the particle number tends to infinity, for a suitable scaling of the interactions. Furthermore, explicit rates for the convergence are obtained in Besov and Triebel–Lizorkin spaces. Our proof is based on the Itô–Wentzell–Kunita formula for Young integral.