Existence of martingale solutions for stochastic flocking models with local alignment

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 F, Smale S (IEEE Transactions on Automatic Control 52(5):852–862, 2007), Cucker F, Smale S (Japanese Journal of Mathematics 2(1):197–227, 2007) and Motsch S, Tadmor E (Journal of Statistical Physics 144(5):923–947, 2011) 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 in Karper T, Mellet A, Trivisa K (Springer Proceedings in Mathematics & Statistics, 2012), Karper T, Mellet A, Trivisa K (SIAM Journal on Mathematical Analysis 45(1):215–243, 2013) in the deterministic case.