  We establish the geometric ergodicity of the preconditioned Hamiltonian Monte
Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as
developed in [Beskos et al., Stochastic Process. Appl., 2011]. This algorithm
can be used as a basis to sample from certain classes of target measures which
are absolutely continuous with respect to a Gaussian measure. Our work
addresses an open question posed in [Beskos et al., Stochastic Process. Appl.,
2011], and provides an alternative to a recent proof based on exact coupling
techniques given in arXiv:1909.07962. The approach here establishes convergence
in a suitable Wasserstein distance by using the weak Harris theorem together
with a generalized coupling argument. We also show that a law of large numbers
and central limit theorem can be derived as a consequence of our main
convergence result. Moreover, our approach yields a novel proof of mixing rates
for the classical finite-dimensional HMC algorithm. As such, the methodology we
develop provides a flexible framework to tackle the rigorous convergence of
other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope
of our result includes certain measures that arise in the Bayesian approach to
inverse PDE problems, cf. [Stuart, Acta Numer., 2010]. Particularly, we verify
all of the required assumptions for a certain class of inverse problems
involving the recovery of a divergence free vector field from a passive scalar,
arXiv:1808.01084v3.
