Geometric ergodicity of two-dimensional Hamiltonian systems with a Lennard–Jones-like repulsive potential

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

“…This is done by constructing an explicit Lyapunov function. The class of admissible potentials, defined below in Section 2, is comparable to those considered in [2,9], but the results are in line with those proven in [3] and hence stronger. In particular, although we do not treat potentials that are merely weakly differentiable on the set where U < 1, our convergence results allow for the analysis of numerical methods used to simulate molecular dynamics or sample from the density using Monte Carlo methods.…”

“…One then solves an equation like (3.5), but with H replaced by the correct dominant operator along different routes to H D 1 when q 0. For further details in the case (3.6), see [3].…”

“…While these ideas are appealing, the scalings used in [3,13] always reduce the problem to studying the deterministic Hamiltonian dynamics at infinity (high H ).…”

“…To do so, we perturb off of our initial guess V 0 D H to produce a Lyapunov function V 0 of the form To see how we should pick , we start by analyzing the dynamics when p is bounded and U.q/ is large. By combining the ideas of Section 3.1 with those in the works [3,17], we will scale the infinitesimal generator L while keeping only the dominant terms. Using this reduced infinitesimal generator will yield a simple PDE for the perturbation so that (3.10) is satisfied.…”

“…In contrast to these works, the authors in [3] establish the existence and exponential convergence to starting from arbitrary initial conditions in an appropriate weighted total variation distance (see [14,20]). However, the setting of this paper was limited to a single particle in dimension d D 1 interacting with the origin via a Lennard-Jones-like potential.…”