Wasserstein stability estimates for covariance-preconditioned Fokker-Planck equations

We study the convergence to equilibrium of the mean field PDE associated with the derivative-free methodologies for solving inverse problems. We show stability estimates in the euclidean Wasserstein distance for the mean field PDE by using optimal transport arguments. As a consequence, this recovers the known convergence towards equilibrium estimates in the case of a linear forward model.