From Stochastic Hamiltonian Systems to Stochastic Compressible Euler Equation

TL;DR

This work analyzes a stochastic Hamiltonian particle system with long-range interactions and proves that the empirical measures for position and velocity converge to the stochastic compressible Euler equations as $N\to\infty$, with convergence quantified in Sobolev-type norms. The authors derive a macroscopic stochastic Euler system by applying the Ito–Kunita–Wentzell framework, commutator estimates, and Taylor expansions to the particle dynamics, culminating in a rigorous mean-field limit result in the presence of common noise. A key contribution is a quantitative rate $N^{-\beta/d}$ for the convergence of the mollified empirical measures to the macroscopic density and momentum, established up to stopping times and in appropriate dual Sobolev spaces. This work provides the first macroscopic derivation of the stochastic compressible Euler equations in this moderately interacting, long-range setting and advances the understanding of stochastic mean-field limits with shared randomness.

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.

Theorems & Definitions (12)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.5
• Lemma 3.1
• proof
• Theorem 3.2
• Remark 3.3
• proof