Wasserstein bounds for invariant measures in the Smoluchowski-Kramers approximation with stable noise

Abstract

We study invariant position distributions in the Smoluchowski-Kramers approximation for Langevin equations driven by rotationally symmetric \(α\)-stable noise, with \(α\in(1,2)\). Under global Lipschitz continuity and dissipativity of the drift, we establish a Wasserstein-1 bound of order \(m^{1/α}\) between the invariant position distribution of the underdamped system and the invariant distribution of its overdamped limit, where $m>0$ is the mass parameter. The proof combines a corrected-position test function with a regularized Stein equation. Cancellation of the transport and friction terms reduces the comparison to a first-moment estimate for the stationary velocity.