For diffusions on Riemannian manifolds with boundary, the paper establishes a quantitative space-time divergence lemma and proves that randomized Hamiltonian Monte Carlo and Langevin dynamics achieve the optimal square-root reduction in relaxation time.
Time averages for kinetic Fokker-Planck equations
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abstract
We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincar\'e and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques.
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Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary
For diffusions on Riemannian manifolds with boundary, the paper establishes a quantitative space-time divergence lemma and proves that randomized Hamiltonian Monte Carlo and Langevin dynamics achieve the optimal square-root reduction in relaxation time.