Gamma Calculus Beyond Villani and Explicit Convergence Estimates for Langevin Dynamics with Singular Potentials

Baudoin, Fabrice; Gordina, Maria; Herzog, David P.

doi:10.1007/s00205-021-01664-1

Cited by 22 publications

(27 citation statements)

References 21 publications

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“…The same is necessary for the complementary result [8], which uses a Lyapunov function approach instead. For further complementary results using the Lyapunov function approach see [9,10] and references therein. In contrast, the earlier mentioned framework by Grothaus and Wang is based on weak Poincaré inequalities introduced by Wang and Röckner in [11], which exist under very weak conditions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincaré Inequalities

Bertram¹,

Grothaus²

2021

Preprint

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We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the L 2convergence rate by using weak Poincaré inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and of weak solutions to the corresponding degenerate stochastic differential equation with multiplicative noise.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincaré Inequalities

Bertram¹,

Grothaus²

2021

Preprint

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“…Although explicit estimates were given in [2], the choice of the weighted norm and constants do not scale well with respect to γ, and thus the estimates do not agree with the predicted exponential convergence rate, which should scale as λ ∝ min{γ, γ −1 }. This is indeed exactly the case for harmonic potentials [39] or for systems on a torus with U = 0 [34], and was also found in various hypocoercive estimates for more general potentials [13,20,31].…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, explicit estimates on the convergence rate were not given, especially as they depend on key parameters of the dynamics, e.g. the friction γ > 0 or the dimension d. This motivated the work [2] which aimed at getting an explicit dependence of the rate on the dimension d, by making use of a certain weighted H 1 topology where the weight satisfies a Lyapunov-type condition. See also [8] for a related work.…”

Section: Introductionmentioning

confidence: 99%

Weighted $L^2$-contractivity of Langevin dynamics with singular potentials

Camrud,

Herzog,

Stoltz

et al. 2021

Preprint

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Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by F. Hérau and developped by Dolbeault, Mouhot and Schmeiser, we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2 (dµ) and L 2 (W * dµ), where µ denotes the invariant probability measure and W * is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min(γ, γ −1 ). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

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“…in [2] and [5]. Singular potentials could be treated also by Lyapunov techniques, see [18] and [9]. Recently, corresponding dynamics and their hypocoercive behavior were studied on abstract smooth manifolds, see [13] or with multiplicative noise, see [10].…”

Section: Introductionmentioning

confidence: 99%

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

Eisenhuth¹,

Grothaus²

2021

Preprint

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The aim of the article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of their corresponding transition semigroups. More generally, we analyze infinite-dimensional nonlinear stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct µ Φ -standard right processes with infinite lifetime and weakly continuous paths providing weak solutions to infinite-dimensional Langevin dynamics with invariant measure µ Φ . The second part deals with the general abstract Hilbert space hypocoercivity method, first described by Dolbeaut, Mouhout and Schmeiser and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. In order to apply the method to infinite-dimensional Langevin dynamics we use an essential m-dissipativity statement for infinite-dimensional Ornstein-Uhlenbeck operators, perturbed by the gradient of a potential, with possible unbounded diffusion operators as coefficients and corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic and Poincaré inequalities for measures with densities w.r.t. infinitedimensional non-degenerate Gaussian measures are substantial. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of the µ Φ -standard right process enables us to show an L 2exponential ergodic result for the weak solution. In the end we apply our results to explicit infinite-dimensional degenerate diffusion equations.

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Gamma Calculus Beyond Villani and Explicit Convergence Estimates for Langevin Dynamics with Singular Potentials

Cited by 22 publications

References 21 publications

Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincaré Inequalities

Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincaré Inequalities

Weighted $L^2$-contractivity of Langevin dynamics with singular potentials

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

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