[Paper Review] Weak and strong mean-field limits for stochastic Cucker-Smale particle systems

Angelo Rosello|arXiv (Cornell University)|May 7, 2019

Geometric Analysis and Curvature Flows15 references4 citations

TL;DR

This paper establishes weak and strong mean-field limits for stochastic Cucker-Smale particle systems with sublinear, locally Lipschitz interaction kernels and combined common and individual Stratonovich noise. Using tightness in Wasserstein spaces and stochastic characteristics, it proves weak convergence to a stochastic PDE and strong $L^p(\Omega)$ convergence under bounded diffusion, enabling propagation of chaos for perturbed Cucker-Smale models.

ABSTRACT

We consider a particle system with a mean-field-type interaction perturbed by some common and individual noises. When the interacting kernels are sublinear and only locally Lipschitz-continuous, relying on arguments based on the tightness of random measures in Wasserstein spaces, we are able to construct a weak solution of the corresponding limiting SPDE. In a setup where the diffusion coefficient on the environmental noise is bounded, this weak convergence can be turned into a strong L^p($\\Omega$) convergence and the propagation of chaos for the particle system can be established. The systems considered include perturbations of the Cucker-Smale model for collective motion.

Motivation & Objective

• To extend mean-field limit results to stochastic Cucker-Smale systems with sublinear, locally Lipschitz interaction kernels and combined common and individual noise.
• To establish weak convergence of the empirical measure to a solution of a limiting stochastic PDE in Stratonovich form.
• To strengthen weak convergence to strong $L^p(\Omega)$ convergence under bounded diffusion coefficients.
• To prove propagation of chaos for the particle system under the derived convergence regime.
• To demonstrate the physical relevance of Stratonovich integration by preserving conservative structure in the limiting SPDE.

Proposed method

• Uses tightness of random measures in Wasserstein spaces to establish weak convergence of the empirical measure as $N \to \infty$.
• Applies stochastic characteristics to analyze the flow of particles and derive estimates under Stratonovich noise.
• Imposes conditions $\sum_k \|\phi_k\|_{\infty}^2 < \infty$ and $\sum_k \|\phi_k\|_{\text{lip}}^2 < \infty$ on noise coefficients to control irregularity.
• Employs stopping times $\tau_R^N$ to control moments and localize the dynamics for $L^p$ estimates.
• Uses symmetry of initial conditions and conditional independence to derive propagation of chaos via convergence of empirical measures.
• Relies on the Itô-Stratonovich correction and transport structure to preserve the conservative form of the limiting SPDE.

Experimental results

Research questions

• RQ1 Can weak mean-field limits be established for stochastic Cucker-Smale systems with sublinear, locally Lipschitz interaction kernels and combined noise?
• RQ2 Does strong $L^p(\Omega)$ convergence of the empirical measure hold when the diffusion coefficient on environmental noise is bounded?
• RQ3 How does Stratonovich noise preserve the conservative structure in the limiting SPDE compared to Itô form?
• RQ4 Can propagation of chaos be rigorously established under the derived strong convergence regime?
• RQ5 What conditions on the noise coefficients ensure convergence of the particle system to the limiting SPDE in law and in $L^p(\Omega)$?

Key findings

• Weak convergence of the empirical measure $\mu^N$ to a weak solution of the limiting SPDE is established via tightness in Wasserstein spaces.
• Under bounded diffusion coefficients, the weak convergence strengthens to strong $L^p(\Omega)$ convergence of $\mu^N$ to $\mu$.
• The limiting SPDE takes the conservative Stratonovich form $d\mu_t + v\cdot\nabla_x\mu_t dt + \nabla_v\cdot(F[\mu]\mu_t)dt + \sum_k \nabla_v\cdot(F_k[\mu]\mu_t)\circ d\beta_t^k = 0$, preserving physical structure.
• Propagation of chaos holds: the finite-dimensional distributions of the particle system converge to i.i.d. copies of the limiting process.
• The conditional law $\mathcal{L}(X^i | \mathcal{F}_T^\beta)$ is shown to be almost surely equal to $\mu$, confirming the mean-field approximation.
• Convergence $X^{i,N} \to X^i$ in $L^p(\Omega; \mathcal{C})$ is proven, confirming pathwise convergence of individual trajectories.