Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions

We establish the geometric ergodicity of preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This can be used a basis to sample from certain classes target measures which are absolutely continuous with respect Gaussian measure. Our work addresses open question posed (2011), and provides alternative recent proof based exact coupling techniques given Bou-Rabee Eberle (Two-scale for infinite dimensions , 2019). The approach here establishes convergence suitable Wasserstein distance by using weak Harris theorem together generalized argument. also show that law large numbers central limit derived consequence our main result. Moreover, yields novel mixing rates classical finite-dimensional HMC algorithm. As such, methodology we develop flexible framework tackle rigorous other Markov Chain algorithms. Additionally, scope result includes arise Bayesian inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, verify all required assumptions class problems involving recovery divergence free vector field passive scalar, Borggaard (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020).