  We describe a new MCMC method optimized for the sampling of probability
measures on Hilbert space which have a density with respect to a Gaussian; such
measures arise in the Bayesian approach to inverse problems, and in conditioned
diffusions. Our algorithm is based on two key design principles: (i) algorithms
which are well-defined in infinite dimensions result in methods which do not
suffer from the curse of dimensionality when they are applied to approximations
of the infinite dimensional target measure on $\bbR^N$; (ii) non-reversible
algorithms can have better mixing properties compared to their reversible
counterparts. The method we introduce is based on the hybrid Monte Carlo
algorithm, tailored to incorporate these two design principles. The main result
of this paper states that the new algorithm, appropriately rescaled, converges
weakly to a second order Langevin diffusion on Hilbert space; as a consequence
the algorithm explores the approximate target measures on $\bbR^N$ in a number
of steps which is independent of $N$. We also present the underlying theory for
the limiting non-reversible diffusion on Hilbert space, including
characterization of the invariant measure, and we describe numerical
simulations demonstrating that the proposed method has favourable mixing
properties as an MCMC algorithm.
