Special CMX Seminar

We study the Lyapunov convergence analysis for degenerate and non-reversible stochastic differential equations (SDEs). We apply the Lyapunov method to the Fokker–Planck equation, in which the Lyapunov functional is chosen as a weighted relative Fisher information functional. We derive a structure condition and formulate the Lyapunov constant explicitly. Given the positive Lyapunov constant, we prove the exponential convergence result for the probability density function towards its invariant distribution in the L1 norm. Several examples are presented: underdamped Langevin dynamics with variable diffusion matrices, quantum SDEs in Lie groups (Heisenberg group, displacement group, and Martinet sub-Riemannian structure), three oscillator chain models with nearest-neighbor couplings, and underdamped mean field Langevin dynamics (weakly self-consistent Vlasov–Fokker–Planck equations). If time is allowable, some extensions will be discussed on the time-inhomogeneous SDEs.