[Paper Review] Entropic multipliers method for langevin diffusion and weighted log sobolev inequalities

Patrick Cattiaux, Arnaud Guillin+2 more|arXiv (Cornell University)|Aug 3, 2017

Markov Chains and Monte Carlo Methods17 references3 citations

TL;DR

This paper introduces an entropic multipliers method to establish hypocoercive convergence to equilibrium for the Langevin diffusion without requiring bounded Hessian of the potential, using a generalized weighted logarithmic Sobolev inequality. The key contribution is a tractable Lyapunov-type condition that ensures entropic decay, removing prior assumptions on the potential's curvature and extending convergence results to unbounded Hessians.

ABSTRACT

In his work about hypocercivity, Villani [18] considers in particular convergence to equilibrium for the kinetic Langevin process. While his convergence results in L 2 are given in a quite general setting, convergence in entropy requires some boundedness condition on the Hessian of the Hamiltonian. We will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the Langevin diffusion. Our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality. As a byproduct, we also give tractable conditions for this functional inequality, which is a particular instance of a weighted logarithmic Sobolev inequality, to hold.

Motivation & Objective

• To remove the bounded Hessian assumption on the potential U in entropic hypocoercivity results for Langevin diffusion.
• To develop a generalized multipliers method applicable to entropy decay, extending the classical L2 hypocoercivity framework.
• To derive tractable sufficient conditions for weighted logarithmic Sobolev inequalities to hold, particularly in the context of kinetic Fokker-Planck dynamics.
• To establish exponential entropic convergence for the Langevin process under weaker regularity assumptions on the potential.

Proposed method

• Introduces an entropic generalization of the multipliers method, adapting the hypocoercivity framework to relative entropy instead of L2 norms.
• Uses a weighted logarithmic Sobolev inequality as a functional inequality tool, with weights derived from the potential and its Hessian.
• Applies the Bakry-Émery criterion and Holley-Stroock perturbation argument to establish convexity and log-Sobolev properties for transformed measures.
• Employs Lyapunov functions and super-Poincaré inequalities to control the tail behavior of the invariant measure and derive entropy decay.
• Constructs a family of auxiliary functions φ and W to localize and control the dynamics in velocity space, particularly for large |y|.
• Derives a super-Poincaré inequality via integration by parts and parameter optimization, leading to a F = ln₊¹ᐟ²-Sobolev inequality.

Experimental results

Research questions

• RQ1 Can entropic hypocoercivity for the Langevin diffusion be established without assuming bounded Hessian of the potential U?
• RQ2 What are sufficient conditions on the potential U that ensure a weighted logarithmic Sobolev inequality holds?
• RQ3 How can the multipliers method be generalized from L2 to entropy-based convergence frameworks?
• RQ4 Can Lyapunov-type conditions be constructed to control the entropy decay in kinetic diffusions with unbounded curvature?
• RQ5 What is the role of the weight function H⁻²ᶿ in the weighted functional inequality and how does it affect convergence rates?

Key findings

• The paper establishes exponential convergence to equilibrium in relative entropy for the Langevin diffusion under a Lyapunov-type condition, without requiring bounded Hessian of the potential.
• A new weighted logarithmic Sobolev inequality is derived, which is equivalent to a super-Poincaré inequality with a specific weight involving H⁻²ᶿ.
• The method enables the removal of the bounded Hessian assumption present in Villani’s original entropic hypocoercivity results.
• A F = ln₊¹ᐟ²-Sobolev inequality is obtained, which implies the desired entropy decay and is stronger than the unweighted version.
• The proof relies on constructing a Lyapunov function W(y) = e^{|y|²/4} and using it to control the tail behavior in velocity space.
• The final inequality is shown to be equivalent to a functional inequality involving ψ(f) = ln¹ᐟ²(e + f)/f, which controls the entropy via a weighted Dirichlet form.