Non-reversible Lifts Of Reversible Diffusion Processes And Relaxation Times

We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. For example, (kinetic) Langevin dynamics and randomised Hamiltonian Monte Carlo are lifts of overdamped Langevin dynamics. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincaré inequalities can be rephrased in the language of lifts and how it can be applied to find optimal lifts.