Optimality of convergence rates for weakly confined kinetic Fokker–Planck equations

Establish whether the convergence rates obtained in Theorem 1.5 for the non-factorised kinetic Fokker–Planck equation with potential V(x)=\langle x\rangle^\alpha/\alpha, 0<\alpha\leq 1, and velocity equilibrium parameter \beta\geq 2 are optimal.

Background

Theorem 1.5 establishes stretched-exponential convergence to a steady state for the non-factorised kinetic Fokker–Planck equation under weak spatial confinement. The authors state that the rates in the regime \beta\geq 2 and for arbitrary \alpha are believed to be optimal, but they do not prove optimality. Thus, determining matching lower bounds or otherwise confirming sharpness remains unresolved.