Hegselmann–Krause model with environmental noise

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.