On Explicit $$L^2$$-Convergence Rate Estimate for Underdamped Langevin Dynamics

Exponential Stability and Hypoelliptic Regularization for the Kinetic Fokker–Planck Equation with Confining Potential

“… Most of the previous works on the exponential convergence as (e.g. [ 9 , 10 , 14 , 15 , 20 , 31 , 32 ]) used the growth condition ( 4 ) to get some weighted Poincaré type inequalities (see [ 31 , Lemma A.24]), which are crucial in these works—and additional to the Poincaré inequality ( 5 ). Our technique is rather different, based on construction of appropriate state dependent matrices and state dependent matrix inequalities so that the (modified) dissipation functional (see ( 20 ) below) decays exponentially.…”

“…By using hypoelliptic methods, Villani proved exponential convergence results in [ 31 , Theorem 35] and in [ 31 , Theorem 37]. The main conditions in Villani’s theorems above, as well as in [ 9 , 10 , 14 , 15 , 20 , 32 ], are the validity of the Poincaré inequality ( 5 ) and the criterion where denotes the Frobenius norm of …”

Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises

“…In particular, as an important application, the L 2 -exponential ergodicity of the degenerate spherical velocity Langevin equation was handled in [15]. See [10] for the recent study on more refined explicit estimates of exponential decay rate of underdamped Langevin dynamics in the L 2 -distance. Meanwhile, the authors in [16] went a step further to generalize the DMS general framework and to tackle the L 2 -algebraic ergodicity of (1.4).…”

A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions

“…In particular, in [24], the authors analyze strong solutions to linear kinetic equations in spatially bounded domains using Hermite polynomials and obtain L 2 error estimates for the approximation which depends on the regularity of the solution. However, in the current paper, we obtain explicit H k error estimates for Galerkin approximations of weak solutions to the kinetic Fokker-Planck equation that only depend on the regularity of the initial data and source term, see (7).…”

“…Different from the approaches mentioned above, Albritton et al [2] proved the existence of weak solutions in T d and exponential decay in an L 2 -setting using a variational approach based on energy methods. Cao et al [7] extended the results of [2] to the whole space and developed a new method for determining the rate of decay to equilibrium in the L 2 norm. Their approach involves expressing the estimate in terms of the Poincaré constant of the underlying space.…”