Hegselmann--Krause model with environmental noise

Abstract: We study a continuous-time version of the Hegselmann-Krause model describing the opinion dynamics of interacting agents subject to random perturbations. Mathematically speaking, the opinion of agents is modelled by an interacting particle system with a non-Lipschitz continuous interaction force, perturbed by idiosyncratic and environmental noises. Sending the number of agents to infinity, we derive a McKean-Vlasov stochastic differential equation as the limiting dynamic, by establishing propagation of chaos for regularized versions of the noisy opinion dynamics. To that end, we prove the existence of a unique strong solution to the McKean-Vlasov stochastic differential equation as well as well-posedness of the associated non-local, non-linear stochastic Fokker-Planck equation.