On gradient structures for Markov chains and the passage to Wasserstein gradient flows

DOI10.3934/nhm.2015.10.233zbMath1372.35124OpenAlexW2164487423WikidataQ59901591 ScholiaQ59901591MaRDI QIDQ258483

Matthias Liero, Karoline Disser

Publication date: 10 March 2016

Published in: Networks and Heterogeneous Media (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.3934/nhm.2015.10.233

zbMATH Keywords

Markov chainsdiscrete gradient flow structuresentropy/entropy-dissipation formulationevolutionary \(\Gamma\)-convergenceWasserstein gradient flow

Mathematics Subject Classification ID

Variational methods for problems in mechanics (70G75) Initial-boundary value problems for second-order parabolic equations (35K20) Continuous-time Markov processes on discrete state spaces (60J27) Discrete approximations in optimal control (49M25) General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations (37L05) Second-order parabolic equations (35K10) Irreversible thermodynamics, including Onsager-Machlup theory (82B35) Limit theorems in probability theory (60F99) Finite volume methods for initial value and initial-boundary value problems involving PDEs (65M08)

Related Items (15)

Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator ⋮ Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities ⋮ Modeling of chemical reaction systems with detailed balance using gradient structures ⋮ Evolutionary $\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker--Planck Equations in Multiple Dimensions ⋮ Homogenisation of one-dimensional discrete optimal transport ⋮ Homogenisation of dynamical optimal transport on periodic graphs ⋮ Hidden Dissipation and Convexity for Kimura Equations ⋮ Cosh gradient systems and tilting ⋮ Diffusive Limit of Random Walks on Tessellations via Generalized Gradient Flows ⋮ Large time behaviors of upwind schemes and $B$-schemes for Fokker-Planck equations on $\mathbb {R}$ by jump processes ⋮ Scaling Limits of Discrete Optimal Transport ⋮ On microscopic origins of generalized gradient structures ⋮ Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit ⋮ Computation of optimal transport on discrete metric measure spaces ⋮ Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

Cites Work

• Unnamed Item
• Unnamed Item
• Passing from bulk to bulk-surface evolution in the Allen-Cahn equation
• Geodesic convexity of the relative entropy in reversible Markov chains
• Gradient flow structures for discrete porous medium equations
• Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction
• A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme
• A numerical method for mass conservative coupling between fluid flow and solute transport
• From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage
• Gradient flows of the entropy for finite Markov chains
• Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory
• Ricci curvature of finite Markov chains via convexity of the entropy
• Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions
• Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes.
• A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
• Fokker-Planck equations for a free energy functional or Markov process on a graph
• \(\Gamma\)-limits and relaxations for rate-independent evolutionary problems
• THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
• Gromov--Hausdorff Convergence of Discrete Transportation Metrics
• Charge Transport in Semiconductors and a Finite Volume Scheme
• A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems
• Finite Volumes for Complex Applications VI Problems & Perspectives
• Weighted energy-dissipation functionals for gradient flows
• A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors
• A finite volume method for density driven flows in porous media
• The Variational Formulation of the Fokker--Planck Equation
• Gamma-convergence of gradient flows with applications to Ginzburg-Landau
• Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation
• Gradient structures and geodesic convexity for reaction–diffusion systems
• On the heat flow on metric measure spaces: existence, uniqueness and stability

This page was built for publication: On gradient structures for Markov chains and the passage to Wasserstein gradient flows