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Quasi-stationary distribution for Hamiltonian dynamics with singular potentials

Abstract

In this work, we prove the existence and the uniqueness of a quasi-stationary distribution for hypoelliptic Hamiltonian dynamics for a system of N particles in \({\mathbb {R}}^d\) interacting with Lennard-Jones type potentials or with repulsive Coulomb potentials.

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Fig. 1

Notes

  1. Existence and uniqueness will be proved later, under additional assumptions on \({\mathsf {V}}\), see Propositions 2.3 and 3.1.

  2. Indeed, on \(\{\tau =+\infty \}\), for all \(t>0\), there exists \(r>0\) such that \(t< \tau _{r}\). Therefore, for all \(s\in [0,t]\), \(X_{s}\in \{ \mathsf {H}_1< r\}\subset {\mathcal {S}}\).

  3. Eq. (2.3) also ensures that \({\mathsf {V}}_{1,{{\textbf { I}}}}\) is lower bounded (see (2.5)), and Lemma A.1 in [30] holds when (HLJ) is satisfied (see Remark 2.6).

  4. Recall that pathwise uniqueness and weak existence imply uniqueness in law, and thus uniqueness of the martingale problem (see for instance [31, Theorems 18.7 and 18.14]).

  5. See more precisely the the computations for the terms \({\mathsf {p}}\cdot \nabla _{\mathsf {q}}\Psi \) and \(-\nabla _{{\mathsf {q}}}U \cdot \nabla _{{\mathsf {p}}}\Psi \) there.

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Acknowledgements

B.N. is supported by the grant IA20Nectoux from the Projet I-SITE Clermont CAP 20-25.

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Correspondence to Boris Nectoux.

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Guillin, A., Nectoux, B. & Wu, L. Quasi-stationary distribution for Hamiltonian dynamics with singular potentials. Probab. Theory Relat. Fields 185, 921–959 (2023). https://doi.org/10.1007/s00440-022-01154-9

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  • DOI: https://doi.org/10.1007/s00440-022-01154-9

Keywords

  • Quasi-stationary distribution
  • Lyapunov functions
  • Hypoelliptic diffusions
  • Molecular dynamics
  • Lennard-Jones potential
  • Coulomb potential

Mathematics Subject Classification

  • 37A60
  • 60B10
  • 60F99
  • 60J60
  • 37A30