On L² convergence of the Hamiltonian Monte Carlo

We represent the abstract Hamiltonian (Hybrid) Monte Carlo (HMC) algorithm as iterations of an operator on densities in a Hilbert space, and recognize two invariant properties of Hamiltonian motion sufficient for convergence. Under a mild coverage assumption, we present a proof of strong convergence of the algorithm to the target density. The proof relies on the self-adjointness of the operator, and we extend the result to the general case of the motions beyond Hamiltonian ones acting on a finite dimensional space, to the motions acting an abstract space equipped with a reference measure, as long as they satisfy the two sufficient properties. For standard Hamiltonian motion, the convergence is also geometric in the case when the target density satisfies a log-convexity condition.