Hey!

I do realise that my question off-topic for Stan as it is purely about theoretical HMC. Still, I thought this was the best place to try to get first help on this. If it is improper here, let me know and/or close the question.

I am struggling to understand why the Markov chain Hamiltonian Monte Carlo produces is positive recurrent. From my understanding any MCMC algorithms needs to produce a chain that is irreducible, positive recurrent and aperiodic so that the chain converges to its stationary distribution and with e.g. the Detailed Balance condition it is ensured that that stationary distribution is the target distribution we specified via potential energy.

I see that HMC produces a chain that is irreducible (because any momentum p can be drawn from the maxwell-boltzman/normal distribution) and aperiodic (because it is always possible to reject a proposal and thus every state has period 1). But with assuring positive recurrence, I donâ€™t find a way. At first I turned to the introduction to HMC by Micheal Betancourt, but didnâ€™t find anything. Now I found the paper â€śOn the convergence of Hamiltonian Monte Carloâ€ť by Alain Durmus, Ă‰ric Moulines, and Eero Saksman on arxiv (Link: https://arxiv.org/pdf/1705.00166) and the arguments should suffice but are pretty technical.

I was wondering if there is a simpler argument (maybe following from other properties of MCMC/HMC) for positive recurrence and thought that if somewhere, then here is the best place to start.