Variational approach has been shown recently to be particularly efficient to prove convergence to equilibrium of irreversible processes such as underdamped Langevin process or sampler based on piecewise deterministic Markov process. By recasting the problem on the concept of lift and collapse process, a general setting is here provided. To do so a general divergence lemma is proved based only on Dirichlet forms under the assumption of a purely discrete spectrum of the collapse process, enabling to establish a general Flow Poincaré inequality necessary for the control of the dissipation of the L2-norm of the lift transition semigroup. Examples of application are given such as to the underdamped Langevin process, but also Markov sampler such as the Forward or the Zig-Zag process where the collapse process is the usual overdamped Langevin process. The case of a pair of Run-and-Tumble particles with jamming on the 1D torus is also considered, whose collapse is a sticky Brownian motion on an interval, and exponential convergence to equilibrium with sharp rates is established.
References:
– Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat, andMatthew Novack. Variational methods for the kinetic fokker-planck equation.arXiv preprint arXiv:1902.04037, to appear in Annals of PDE, 2019.
– Yu Cao, Jianfeng Lu, and Lihan Wang. On explicit l 2-convergence rate estimate for underdamped langevin dynamics. Archive for Rational Mechanics and Analysis, 247(5):90, 2023.
– Andreas Eberle and Francis Lörler. Non-reversible lifts of reversible diffusion processes and relaxation times.arXiv preprint arXiv:2402.05041,2024.
– Jianfeng Lu and Lihan Wang. On explicit l 2-convergence rate estimatefor piecewise deterministic markov processes in mcmc algorithms.TheAnnals of Applied Probability, 32(2):1333–1361, 2022
In collaboration with Andreas Eberle, Léo Hahn, Francis Lörler, Manon Michel