Second Order Ensemble Langevin Method for Sampling and Inverse Problems

The paper’s Proposition 5.1 states and proves the closed ODEs for the mean and block covariances, classifies all steady states (unique positive-definite equilibrium and a family of degenerate equilibria), and establishes local exponential stability in B-whitened variables with a rate depending only on γ, independent of B and c. The candidate solution reproduces the same moment equations, obtains the same steady-state structure via the projection X characterization, performs the same whitening, and derives the same cubic characteristic polynomial λ^3 + 3γ λ^2 + (2γ^2 + 6) λ + 4γ for the 3×3 covariance block. It adds a Routh–Hurwitz check and a standard small-data bootstrap to articulate the open basin and uniform-in-(B,c) rate. All core steps align with the paper’s derivations and claims, including the block structure of the Jacobian and the independence of B and c after whitening. Hence both are correct and essentially the same proof, with the model supplying slightly more explicit stability folklore details. See the statement of (5.2) and Proposition 5.1 and its proof in Section 8.4, including the whitened system (8.1) and the cubic eigenvalue relation, as well as the independence from B and c and hyperbolicity claims in the paper.