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 that are presented by Garbuno-Inigo et al (2020 SIAM J. Appl. Dyn. Syst. 19 412–41), Herty and Visconti (2018 arXiv: 1811.09387 ). 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 convergence towards equilibrium estimates by Garbuno-Inigo et al (2020 SIAM J. Appl. Dyn. Syst. 19 412–41) in the case of a linear forward model.