HMC, an example of Functional Analysis applied to Algorithms in Data Mining. The convergence in $L^p$

We present a proof of convergence of the Hamiltonian Monte Carlo algorithm in terms of Functional Analysis. We represent the algorithm as an operator on the density functions, and prove the convergence of iterations of this operator in $L^p$, for $1<p<infty$, and strong convergence for $2le p<infty$.