Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential

By Frédéric Hérau, Francis Nier. 2004

350 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/678bcb428e7356afe77b2fdffbfb6e80d39e2b91

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Citations in the corpus

Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, background): “(2) We did not include it in the statement for the sake of clarity but our method of proof can recover the semigroup decay estimate in L(2)(μ) as a consequence of the known decay estimate in H(1)(μ) proved in [14, 16, 24], which provides an alternative argument to those in [8, 9].”
Factorization for non-symmetric operators and exponential H-theorem (2010, neutral): “veloped in the recent works [36, 35], partly inspired from the paper [53]. In [35] the kinetic equation is studied in the whole space with conning potential. This result is also related to the works [54] and [97] on the kinetic Fokker-Planck equation in the whole space with a conning potential. Theo”
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation (2005, methodology): “An example of such interplay is given by the algebraic structure revealed in the beautiful recent series of works by H erau, Nier and Heler [36, 37 ] about the linear kinetic Fokker-Planck equation.”
HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS (2010, result): “). Then solutions of (1.1) with initial data in L2(dµ) decay exponentially to the global equilibrium given by (3.1). The above assumptions are similar to those of [33], which are weaker than those of [21]. Moreover this result is an important improvement compared to [33] as involves an L2 setting ra”
Hypocoercivity (2006, methodology): “Here is a more precise comparison: Helffer and Nier prove exponential convergence under two sets of assumptions: on one hand, [30, Assumption 5.6]; on the other hand, [30, Assumption 5.7] plus a spectral gap condition which is equivalent to (7.5).”
Underdamped Langevin MCMC: A non-asymptotic analysis (2017, background): “The continuous-time process has been studied extensively [12, 13, 14, 15, 16, 17, 18, 19, 20].”
Spectral analysis of semigroups and growth-fragmentation equations (2013, background): “We refer the interested reader to [104] for a pedagogical introduction as well as to [56, 53, 88, 54, 55, 105, 29, 28] for some of the original articles.”
Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex (2011, methodology): “The accurate computations of its exponentially small eigenvalues has connections with various topics such as stochastic analysis ([6,7,14]), kinetic theory ([21,25–28]), the computation of geometric invariants ([3,4]), or differential topology ([8,31,38]).”
Explicit parametrix and local limit theorems for some degenerate diffusion processes (2008, background): “PARAMETRIX FOR SOME DEGENERATE PROCESSES 33[HN04℄ F. Hérau and F. Nier.”
Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus (2006, methodology): “Also another approach has been developed in [25] for the Fokker-Planck model in a confining external field (see also [24]), and Hérau recently generalized the method to the linear relaxation model in [23].”
Density estimates for a random noise propagating through a chain of differential equations (2010, background): “(1.1) describes the dynamics of some stochastic Hamiltonian systems (see e.g. Soize [Soi94] for a general overview or the more specic works by Talay [Tal02] and Hérau and Nier [HN04] for questions of convergence to equilibrium).”
Short and long time behavior of the Fokker-Planck equation in a confining potential and applications (2005, methodology): “1 ([14]-[15]) Operators K and K∗ defined as the closure of (2) with domain C∞ 0 are maximal accretive.”
The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics (2013, methodology): “The interest of this approach is that it allows us to give more explicit rates of convergence than the ones obtained by Lyapunov type approaches, and gives fine results on the structure of the spectrum of Lγ (see in particular Eckmann & Hairer (2003); Hérau & Nier (2004)).”
Parametrix techniques and martingale problems for some degenerate Kolmogorov equations (2010, background): “Let us for instance mention for n = 2 stochastic Hamiltonian systems (see e.g. Soize [Soi94] for a general overview or Talay [Tal02] and H´erau and Nier [HN04] for convergence to equilibrium).”
Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows (2015, background): “[17,18,25,26] and the references therein or the text [47] for an overview of the basic ideas and a general study of hypocoercivity applied to linear operators in Hörmander form.”
Couplings and quantitative contraction rates for Langevin dynamics (2017, background): “Several analytic approaches to convergence to equilibrium for kinetic Fokker-Planck equations have been proposed during the last 15 years [10, 17, 27, 25, 26, 40, 42, 11, 7, 33, 19, 20, 3].”