Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition

Abstract

In this paper, we consider the Cucker-Smale flocking particles which are subject to the same velocitydependent noise, which exhibits a phase change phenomenon occurs bringing the system from a "non flocking" to a "flocking" state as the strength of noises decreases. We rigorously show the stochastic mean-field limit from the many-particle Cucker-Smale system with multiplicative noises to the Vlasov-type stochastic partial differential equation as the number of particles goes to infinity. More precisely, we provide a quantitative error estimate between solutions to the stochastic particle system and measure-valued solutions to the expected limiting stochastic partial differential equation by using the Wasserstein distance. For the limiting equation, we construct global-in-time measure-valued solutions and study the stability and large-time behavior showing the convergence of velocities to their mean exponentially fast almost surely.

The Microscopic Derivation and Well-Posedness of the Stochastic Keller–Segel Equation

“…We refer to Bensoussan et al (2013), Carmona et al (2016), Carmona and Delarue (2018), Coghi and Flandoli (2016) for different models with common noise in the literature. In particular, in a closely related work (Coghi and Flandoli 2016), the authors study the propagation of chaos for an interacting particle system subject to a common environmental noise but with a uniformly Lipschitz continuous potential, and in Choi and Salem (2019), the stochastic mean-field limit of the Cucker-Smale flocking particle system is obtained for a special class of noises. In contrast to the existing literature concerning common noise, the main difficulties in dealing with the proposed stochastic KS models are from the Bessel potential G which entails the singularity of the drift of SDE (1.7) and the KS type nonlinear and nonlocal properties of SPDE (1.3); in particular, the KS type nonlinear term −χ ∇ • ((∇G * ρ t )ρ t ) prevents us from adopting the existing methods in the SPDE literature.…”

Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

“…The propagation of chaos and its consequence, the rigorous proof of the mean-field limit, are questions of nowadays importance in many other problems; for instance, in kinetic models of collective behavior [1,4,6,9,10,11] with or without noise in velocity, see also [12,28] for more general types of equations. However, in most of these applications the singularity of the kernels is much better behaved.…”

Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communications

“…Derivation of the SPDE. In this subsection, following [8], we present a derivation of (1.3) from the C-S system perturbed by a multiplicative noise. To be specific, we begin our discussion with the C-S model [14].…”

“…However, if each W i t is identical to a single Wiener process W , i.e. W i ≡ W t , we can use a propagation of chaos result [8] to obtain that the empirical measure µ N t associated with system (2.2) converges to a measure-valued solution to (1.3).…”

“…At the particle level, the effects of white noise perturbations were discussed in [1,16,23]. Moreover, a rigorous derivation of equation (1.3) as a mean-field limit of the C-S systems with multiplicative noises was recently discussed in [8] based on the propagation of chaos in [10], and a mean-field limit of the C-S systems with another type of stochastic perturbations was also addressed in [35]. However, as far as the authors know, the kinetic C-S equation (1.2) perturbed by a multiplicative white noise has only been addressed in measure spaces such as P 2 , not in other function spaces (e.g.…”