Continuous-time limit of repeated interactions for a system in a confining potential

We study the continuous-time limit of a class of Markov chains coming from the evolution of classical open systems undergoing repeated interactions. This repeated interaction model has been initially developed for dissipative quantum systems in Attal and Pautrat (2006) and was recently set up for the first time in Deschamps (2012) for classical dynamics. It was particularly shown in the latter that this scheme furnishes a new kind of Markovian evolutions based on Hamilton's equations of motion. The system is also proved to evolve in the continuous-time limit with a stochastic differential equation. We here extend the convergence of the evolution of the system to more general dynamics, that is, to more general Hamiltonians and probability measures in the definition of the model. We also present a natural way to directly renormalize the initial Hamiltonian in order to obtain the relevant process in a study of the continuous-time limit. Then, even if Hamilton's equations have no explicit solution in general, we obtain some bounds on the dynamics allowing us to prove the convergence in law of the Markov chain on the system to the solution of a stochastic differential equation, via the infinitesimal generators.