Perelman's \(W\)-entropy for the Fokker-Planck equation over complete Riemannian manifolds

DOI10.1016/j.bulsci.2011.05.003zbMath1230.82039OpenAlexW1991959037WikidataQ115359975 ScholiaQ115359975MaRDI QIDQ645955

Xiang Dong Li

Publication date: 11 November 2011

Published in: Bulletin des Sciences Mathématiques (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.bulsci.2011.05.003

zbMATH Keywords

Fokker-Planck equationPerelman \(W\)-entropy

Mathematics Subject Classification ID

Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics (82C31) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Fokker-Planck equations (35Q84)

Related Items

\(W\)-entropy, super Perelman Ricci flows, and \((K, m)\)-Ricci solitons ⋮ The logarithmic entropy formula for the linear heat equation on Riemannian manifolds ⋮ Hamilton differential Harnack inequality and \(W\)-entropy for Witten Laplacian on Riemannian manifolds ⋮ Monotonicity and rigidity of the \(\mathcal{W}\)-entropy on \(\mathsf{RCD} (0, N)\) spaces ⋮ \(W\)-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds ⋮ Hamilton's Harnack inequality and the \(W\)-entropy formula on complete Riemannian manifolds

Cites Work

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• Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces
• Perelman's entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature
• Fokker-Planck equation with respect to heat measures on loop groups
• Variational approximation for Fokker-Planck equation on Riemannian manifold
• The heat equation on manifolds as a gradient flow in the Wasserstein space
• Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature
• On the parabolic kernel of the Schrödinger operator
• The entropy formula for linar heat equation
• Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
• Convex functionals of probability measures and nonlinear diffusions on manifolds
• Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds
• ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS
• THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
• Uniqueness for Solutions of Fokker–Planck Equations on Infinite Dimensional Spaces
• Harmonic functions on complete riemannian manifolds
• The Variational Formulation of the Fokker--Planck Equation
• Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation