  There is substantial empirical evidence about the success of dynamic
implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler
(NUTS), in many challenging inference problems but theoretical results about
their behavior are scarce. The aim of this paper is to fill this gap. More
precisely, we consider a general class of MCMC algorithms we call dynamic HMC.
We show that this general framework encompasses NUTS as a particular case,
implying the invariance of the target distribution as a by-product. Second, we
establish conditions under which NUTS is irreducible and aperiodic and as a
corrolary ergodic. Under conditions similar to the ones existing for HMC, we
also show that NUTS is geometrically ergodic. Finally, we improve existing
convergence results for HMC showing that this method is ergodic without any
boundedness condition on the stepsize and the number of leapfrog steps, in the
case where the target is a perturbation of a Gaussian distribution.
