Hypocoercivity

By Cédric Villani. 2006

546 citations recorded by the source corpus.

This memoir attempts at a systematic study of convergence to stationary state for certain classes of degenerate diffusive equations, by means of well-chosen Lyapunov functionals. Typical examples are the kinetic Fokker--Planck and Boltzmann equation. Many open problems and possible directions for future research are discussed.

Subfields: Physiology

View Paper: https://www.semanticscholar.org/paper/3244be98bbcc1aa78a9f57a9a0699bab599e231f

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In Physiology

Citations in the corpus

Stochastic Heavy ball (2016, methodology): “Recent works fruitfully exploit this kind of Lyapunov function (see, among others, the kinetic Fokker-Planck equations in [Vil09] and the memory gradient diffusion in [GP14]).”
Enhanced dissipation and Hörmander's hypoellipticity (2021, methodology): “A commonly used method to prove enhanced dissipation [8] is hypocoercivity, whose mathematical framework was developed by Villani in [43].”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, methodology): “This theorem is proved by combining: • The spectral gap property of the Fokker-Planck semigroup which is classically known in the space of self-adjointness L2(μ−1/2) in a spatially homogeneous setting (Poincaré inequality) and has been recently proved in a series of works about “hypocoercivity” in t”
Derivative formula and applications for degenerate diffusion semigroups (2011, background): “[12] where the hypocoercivity and related regularization estimates w.”
Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow (2014, background): “It is pointed out in [8] that the effect can be linked to a form of ‘hypocoercivity’ [54].”
Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows (2015, methodology): “The connection between hypocoercivity and hypoellipticity is the origin of the name of the former [47], however, Theorem 1.”
Semi Log-Concave Markov Diffusions (2013, methodology): “The latter property is due to Otto-Villani [38].”
Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity (2017, background): “This phenomenon also occurs for the kinetic diﬀusion dynamics [Vil09, GM13]) and for the Nesterov accelerated gradient descent [Nes83, CEG09] even though we do not claim that such a clear common point exists be-tween these methods.”
Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions (2022, methodology): “In [Ma+21], following the entropic hypocoercivity approach of [Vil09], Ma et al. consider a Lyapunov functional L which is equivalent to the sum of the KL divergence and the Fisher information, L ( µ γ 2 d ) ≍ KL ( µ γ 2 d ) + FI ( µ γ 2 d ) , which decays exponentially fast in time: there exists a…”
On Learning Rates and Schrödinger Operators (2020, methodology): “For completeness, we show below the original proof of Lemma 5.4 from [Vil09] in detail.”
Underdamped Langevin MCMC: A non-asymptotic analysis (2017, background): “The continuous-time process has been studied extensively [7, 8, 9, 10, 11, 12, 13, 14, 15].”
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré (2007, background): “In addition, in the last version of his work (see [31]) Villani gives some explicit bounds for the constants involved.”
Couplings and quantitative contraction rates for Langevin dynamics (2017, methodology): “Let us now set our results in relation to explicit bounds for convergence to equilibrium of kinetic Fokker-Planck equations that have been obtained in [27, 42, 11] by analytic methods.”
Hypercontractivity and applications for stochastic Hamiltonian systems (2017, background): “SHS)” in probability theory (see [22]), and the distribution density of the solution solves the kinetic Fokker-Planck equation (see [14]).”
Exponential Rate of Convergence to Equilibrium for a Model Describing Fiber Lay-Down Processes (2012, background): “This is a so-called hypocoercivity result as defined in [13].”
Linear inviscid damping and enhanced dissipation for the Kolmogorov flow (2017, methodology): “The first part of this paper is devoted to studying the linear inviscid damping for the linearized Euler equations around the Kolmogorov flow (− cos y, 0), which take in the vorticity formulation: { ∂tω + Lω = 0, ω|t=0 = ω0(x, y),(1.2) where ω = ∂xV 2 − ∂yV 1 is the vorticity and L = u(y)∂x + u′′(y)”