ON THE TREND TO EQUILIBRIUM FOR THE FOKKER-PLANCK EQUATION : AN INTERPLAY BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS

By Peter A. Markowich, Cédric Villani. 2004

207 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/2d78233b0cfe600c2bcabd4c7e0c0dd378140e5c

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Citations in the corpus

Generalization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints (2017, methodology): “The first term is minus Fisher information I(π||γ), which can be upper bounded by −DKL(π||γ) itself using logarithmic Sobolev inequality (Gross, 1975; Markowich and Villani, 2000): DKL(π||γ) ≤ σ(2) 0I(π||γ), for γ = N (0, σ(2) 0I) (51) Let C = β and plug into the log Sobolev inequality, we get:”
Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory (2002, background): “However, all limits of the type Boltzmann → Landau can be put into a unified formalism, first sketched in our work [446], then extended and made more precise in Alexandre and Villani [12].”
Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems (2008, methodology): “Finally, we note that relative entropy and more general logarithmic Sobolev inequalities have been extensively used both to bound the heat kernel (Green’s function) of elliptic operators as well as to study the decay to equilibrium of more general Fokker-Planck type equations, see [15, 35].”
On Learning Rates and Schrödinger Operators (2020, unknown): “Deﬁnition 2.1 (Conﬁning condition [Pav14, MV99]) .”
The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation (2009, background): “Concerning this wide subject, a recent study [36] reveals the various links between the Fokker–Planck equation and differential inequalities.”
Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates (2003, methodology): “The surprising success of this method [1, 11, 12, 9] (see [23] for a tentative user-friendly review on these techniques), which may seem hard to believe at first, can be explained at a heuristical level by the conceptual work of Otto [27], who showed that the relation of the equation (1.”