Existence of martingale solutions for stochastic flocking models with local alignment

TL;DR

This paper proves the existence of martingale solutions for stochastic models of collective motion, extending deterministic results to include randomness and perturbations in flocking models like Cucker-Smale.

Contribution

It introduces a method to construct approximate solutions via regularization and mean-field limits, establishing existence in a stochastic setting for flocking models.

Findings

Existence of martingale solutions for stochastic flocking models.

Extension of deterministic results to stochastic perturbations.

Use of stochastic averaging lemma for compactness in law.

Abstract

We establish the existence of martingale solutions to a class of stochastic conservation equations. The underlying models correspond to random perturbations of kinetic models for collective motion such as the Cucker-Smale and Motsch-Tadmor models. By regularizing the coefficients, we first construct approximate solutions obtained as the mean-field limit of the corresponding particle systems. We then establish the compactness in law of this family of solutions by relying on a stochastic averaging lemma. This extends the results obtained by Karper, Mellet and Trivisa in the deterministic case.