Geometric ergodicity of Langevin dynamics with Coulomb interactions

Lu, Yulong; Mattingly, Jonathan C.

doi:10.1088/1361-6544/ab514a

Cited by 23 publications

(31 citation statements)

References 42 publications

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“…The Lyapunov function (4.13) can be adapted in cases where V has singularities, see [64,85]. We can now deduce our main theorem on the Langevin dynamics since Assumptions 2.5 and 2.6 are readily satisfied, see for instance [86].…”

Section: Large Deviationsmentioning

confidence: 79%

“…Note that constructing a control u ∈ C 0 ([0, T ], X ) may be difficult in general [70]. However, for the overdamped and underdamped Langevin dynamics we are interested in, building such a control turns out to be guenuinely feasible, see [86,105,97,83,85] and references therein. Let us mention that the above two assumptions are standard for proving LDPs [109,115].…”

Section: Large Deviations Of Empirical Measures Of Diffusions In Weighted Topologiesmentioning

confidence: 99%

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Large deviations of empirical measures of diffusions in weighted topologies

Ferré¹,

Stoltz²

2020

Electron. J. Probab.

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We consider large deviations of empirical measures of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) for a precise class of unbounded functions. This provides an analogue to the standard Cramér condition in the context of diffusion processes, which turns out to be related to a spectral gap condition for a Witten-Schrödinger operator. Secondly, we study more precisely the properties of the Donsker-Varadhan rate functional associated with the LDP. We revisit and generalize some standard duality results as well as a more original decomposition of the rate functional with respect to the symmetric and antisymmetric parts of the dynamics. Finally, we apply our results to overdamped and underdamped Langevin dynamics, showing the applicability of our framework for degenerate diffusions in unbounded configuration spaces.

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Section: Large Deviationsmentioning

confidence: 79%

Section: Large Deviations Of Empirical Measures Of Diffusions In Weighted Topologiesmentioning

confidence: 99%

Large deviations of empirical measures of diffusions in weighted topologies

Ferré¹,

Stoltz²

2020

Electron. J. Probab.

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“…Mean-field couplings add a serious difficulty: see [40,57] for recent results based on a probabilistic approach. In presence of a Poisson coupling large time behavior (without rates) of the solutions of (5) has been dealt with in presence of or without an external potential: cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

$$\text {L}^2$$-Hypocoercivity and Large Time Asymptotics of the Linearized Vlasov–Poisson–Fokker–Planck System

Addala

Dolbeault

et al. 2021

J Stat Phys

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This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.

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“…In this dynamics, (X t ) t≥0 stands for a position, while (Y t ) t≥0 represents a momentum variable. Let us mention that the long time convergence of the law of this process towards P n (a difficult problem due to the singularity of the Hamiltonian) can be proved through Lyapunov function techniques, see [38] for a recent account. In practice, the singularity of g also makes the numerical integration of (4.10) difficult, and a Metropolis-Hastings selection rule can be used to stabilize the numerical discretization, see [11] and references therein.…”

Section: Constrained Langevin Dynamicsmentioning

confidence: 99%

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

Chafäı¹,

Ferré²,

Stoltz³

2021

SIAM J. Math. Anal.

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We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an Icontinuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system. Contents 12 4.1. Description of the algorithm 12 4.2. Numerical results 17 Acknowledgements 20 Appendix A.

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Geometric ergodicity of Langevin dynamics with Coulomb interactions

Cited by 23 publications

References 42 publications

Large deviations of empirical measures of diffusions in weighted topologies

Large deviations of empirical measures of diffusions in weighted topologies

$$\text {L}^2$$-Hypocoercivity and Large Time Asymptotics of the Linearized Vlasov–Poisson–Fokker–Planck System

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

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