Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

Abstract

We study Langevin dynamics of N particles on ℝd interacting through a singular repulsive potential, e.g., the well‐known Lennard‐Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. © 2019 Wiley Periodicals, Inc.

Well-posedness and long time behavior of singular Langevin stochastic differential equations

“…Recently, the case of the SDE (1.1) with a singular potential field F has been studied in [3,4,10]. The exponential ergodicity was also obtained by constructing explicit Lyapunov functions, see also the recent work [13] and the references therein. We intend to study the existence and uniqueness of strong solutions as well as the exponential ergodicity of (1.1) under the presence of a singular velocity field G, which destroys the dissipation in the momentum part and makes the classical Lyapunov condition very difficult to check, if possible at all.…”

Geometric ergodicity of Langevin dynamics with Coulomb interactions

“…Proof. The proof of the proposition is the same as the proof of [15,Corollary 5.12] (see also [23,Lemma 2.3]) and hence omitted.…”

“…Recently, the authors of [15] proposed a different idea to construct a Lyapunov function and used it to prove the geometric ergodicity of Langevin dynamics with a larger class of admissible singular potentials U . More specifically, their admissible condition on U states that for any sequence {q k } ⊂ D with U (q k ) → ∞,…”

“…Since our construction of Lyapunov function for the Coulomb system follows closely that of [15], it is worth to briefly review the techniques therein. Their construction relies on the following key observation (which was described earlier in [7]): under the following parabolic scaling…”

“…A typical choice of Ψ suggested by [15] is a function Ψ ∈ C 2 (X ) and Ψ = κ p · ∇U (q) |∇U (q)| 2 on X ∩ {U ≥ R} for some large constant κ > 0 which leads to AΨ = −κ on X ∩ {U ≥ R}. Therefore on X ∩ {U ≥ R}, it holds that (3.9)…”