Contraction in the underdamped regime for generalized Langevin equations with memory

Establish quantitative contraction estimates for the second-order generalized Langevin equation with additive white noise and a Volterra memory term ∫_0^t K(t,s) V_s ds under small friction (the underdamped regime), for strongly confining potentials U(x) with Lipschitz gradients, analogous to the contraction results available for the memoryless Langevin equation via PDE techniques.

Background

The paper proves trajectory-wise stability and contraction for generalized Langevin equations using coupling and Volterra resolvent techniques, primarily under conditions that align with the overdamped regime (large friction). The chosen Lyapunov-type distance and assumptions ensure contraction when γ is sufficiently large.

The authors note that PDE-based hypocoercive methods provide quantitative contraction in the underdamped regime for classical (memoryless) Langevin dynamics, but extending comparable quantitative estimates to generalized Langevin equations that include memory kernels is not yet available. Addressing this would bridge hypocoercivity with non-Markovian memory effects.