From Stochastic Hamiltonian Systems to Stochastic Compressible Euler Equation

TL;DR

This paper demonstrates that a stochastic Hamiltonian particle system converges to stochastic compressible Euler equations as the number of particles grows, providing quantitative measures of the convergence in Sobolev norms.

Contribution

It establishes the rigorous derivation of stochastic compressible Euler equations from a many-particle stochastic Hamiltonian system, including convergence rates.

Findings

Empirical measures converge to stochastic Euler solutions as particle number increases.

Quantitative Sobolev norm estimates of particle-limit distance.

Validation of the mean-field limit for stochastic Hamiltonian systems.

Abstract

We study a stochastic Hamiltonian system of $N$ particles with many particles interacting through a potential whose range is large in comparison with the typical distance between neighbouring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of stochastic compressible Euler equations in the limit as the particle number tends to infinity. Moreover, we quantify the distance between particles and the limit in suitable Sobolev norm.