The Resource Regularity theory for mean field games systems, Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan
Regularity theory for mean field games systems, Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan
Resource Information
The item Regularity theory for mean field games systems, Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.This item is available to borrow from 1 library branch.
Resource Information
The item Regularity theory for mean field games systems, Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.
This item is available to borrow from 1 library branch.
- Summary
- Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields
- Language
- eng
- Extent
- 1 online resource.
- Contents
-
- Preface; Book Outline; Thanks; Bibliographical Notes; Acknowledgments; Contents; 1 Introduction; 1.1 Derivation of MFG Models; 1.1.1 Optimal Control and Hamilton-Jacobi Equations; 1.1.2 Transport Equation; 1.1.3 Mean-Field Models; 1.1.4 Extensions and Additional Problems; 1.1.5 Uniqueness; 2 Explicit Solutions, Special Transformations, and Further Examples; 2.1 Explicit Solutions; 2.2 The Hopf-Cole Transform; 2.3 Gaussian-Quadratic Solutions; 2.4 Interface Formation; 2.5 Bibliographical Notes; 3 Estimates for the Hamilton-Jacobi Equation; 3.1 Comparison Principle; 3.2 Control Theory Bounds
- 3.2.1 Optimal Trajectories3.2.2 Dynamic Programming Principle; 3.2.3 Subdifferentials and Superdifferentials of the Value Function; 3.2.4 Regularity of the Value Function; 3.3 Integral Bernstein Estimate; 3.4 Integral Estimates for HJ Equations; 3.5 Gagliardo-Nirenberg Estimates; 3.6 Bibliographical Notes; 4 Estimates for the Transport and Fokker-Planck Equations; 4.1 Mass Conservation and Positivity of Solutions; 4.2 Regularizing Effects of the Fokker-Planck Equation; 4.3 Fokker-Planck Equation with Singular Initial Conditions; 4.4 Iterative Estimates for the Fokker-Planck Equation
- 4.4.1 Regularity by Estimates on the Divergence of the Drift4.4.2 Polynomial Estimates for the Fokker-Planck Equation, p<∞; 4.4.3 Polynomial Estimates for the Fokker-Planck Equation, p=∞; 4.5 Relative Entropy; 4.6 Weak Solutions; 4.7 Bibliographical Notes; 5 The Nonlinear Adjoint Method; 5.1 Representation of Solutions and Lipschitz Bounds; 5.2 Conserved Quantities; 5.3 The Vanishing Viscosity Convergence Rate; 5.4 Semiconcavity Estimates; 5.5 Lipschitz Regularity for the Heat Equation; 5.6 Irregular Potentials; 5.7 The Hopf-Cole Transform; 5.8 Bibliographical Notes; 6 Estimates for MFGs
- 6.1 Maximum Principle Bounds6.2 First-Order Estimates; 6.3 Additional Estimates for Solutions of the Fokker-Plank Equation; 6.4 Second-Order Estimates; 6.4.1 Stationary Problems; 6.4.2 Time-Dependent Problems; 6.5 Some Consequences of Second-Order Estimates; 6.6 The Evans Method for the Evans-Aronsson Problem; 6.7 An Energy Conservation Principle; 6.8 Porreta's Cross Estimates; 6.9 Bibliographical Notes; 7 A Priori Bounds for Stationary Models; 7.1 The Bernstein Method; 7.2 A MFG with Congestion; 7.3 Logarithmic Nonlinearity; 7.4 Bibliographical Notes
- 8 A Priori Bounds for Time-Dependent Models8.1 Subquadratic Hamiltonians; 8.2 Quadratic Hamiltonians; 8.3 Bibliographical Notes; 9 A Priori Bounds for Models with Singularities; 9.1 Logarithmic Nonlinearities; 9.2 Congestion Models: Local Existence; 9.2.1 Estimates for Arbitrary Terminal Time; 9.2.2 Short-Time Estimates; 9.3 Bibliographical Notes; 10 Non-local Mean-Field Games: Existence; 10.1 First-Order, Non-local Mean-Field Games; 10.2 Second-Order, Non-local Mean-Field Games; 10.3 Bibliographical Notes; 11 Local Mean-Field Games: Existence; 11.1 Bootstrapping Regularity
- Isbn
- 9783319389325
- Label
- Regularity theory for mean field games systems
- Title
- Regularity theory for mean field games systems
- Statement of responsibility
- Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan
- Language
- eng
- Summary
- Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields
- Cataloging source
- YDX
- http://library.link/vocab/creatorName
- Gomes, Diogo A
- Dewey number
- 530.14/4
- Index
- index present
- LC call number
- QC174.85.M43
- Literary form
- non fiction
- Nature of contents
-
- dictionaries
- bibliography
- http://library.link/vocab/relatedWorkOrContributorName
-
- Pimentel, Edgard A.
- Voskanyan, Vardan
- Series statement
- SpringerBriefs in mathematics
- http://library.link/vocab/subjectName
-
- Mean field theory
- Game theory
- Label
- Regularity theory for mean field games systems, Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan
- Bibliography note
- Includes bibliographical references and index
- Carrier category
- online resource
- Carrier category code
-
- cr
- Carrier MARC source
- rdacarrier
- Content category
- text
- Content type code
-
- txt
- Content type MARC source
- rdacontent
- Contents
-
- Preface; Book Outline; Thanks; Bibliographical Notes; Acknowledgments; Contents; 1 Introduction; 1.1 Derivation of MFG Models; 1.1.1 Optimal Control and Hamilton-Jacobi Equations; 1.1.2 Transport Equation; 1.1.3 Mean-Field Models; 1.1.4 Extensions and Additional Problems; 1.1.5 Uniqueness; 2 Explicit Solutions, Special Transformations, and Further Examples; 2.1 Explicit Solutions; 2.2 The Hopf-Cole Transform; 2.3 Gaussian-Quadratic Solutions; 2.4 Interface Formation; 2.5 Bibliographical Notes; 3 Estimates for the Hamilton-Jacobi Equation; 3.1 Comparison Principle; 3.2 Control Theory Bounds
- 3.2.1 Optimal Trajectories3.2.2 Dynamic Programming Principle; 3.2.3 Subdifferentials and Superdifferentials of the Value Function; 3.2.4 Regularity of the Value Function; 3.3 Integral Bernstein Estimate; 3.4 Integral Estimates for HJ Equations; 3.5 Gagliardo-Nirenberg Estimates; 3.6 Bibliographical Notes; 4 Estimates for the Transport and Fokker-Planck Equations; 4.1 Mass Conservation and Positivity of Solutions; 4.2 Regularizing Effects of the Fokker-Planck Equation; 4.3 Fokker-Planck Equation with Singular Initial Conditions; 4.4 Iterative Estimates for the Fokker-Planck Equation
- 4.4.1 Regularity by Estimates on the Divergence of the Drift4.4.2 Polynomial Estimates for the Fokker-Planck Equation, p<∞; 4.4.3 Polynomial Estimates for the Fokker-Planck Equation, p=∞; 4.5 Relative Entropy; 4.6 Weak Solutions; 4.7 Bibliographical Notes; 5 The Nonlinear Adjoint Method; 5.1 Representation of Solutions and Lipschitz Bounds; 5.2 Conserved Quantities; 5.3 The Vanishing Viscosity Convergence Rate; 5.4 Semiconcavity Estimates; 5.5 Lipschitz Regularity for the Heat Equation; 5.6 Irregular Potentials; 5.7 The Hopf-Cole Transform; 5.8 Bibliographical Notes; 6 Estimates for MFGs
- 6.1 Maximum Principle Bounds6.2 First-Order Estimates; 6.3 Additional Estimates for Solutions of the Fokker-Plank Equation; 6.4 Second-Order Estimates; 6.4.1 Stationary Problems; 6.4.2 Time-Dependent Problems; 6.5 Some Consequences of Second-Order Estimates; 6.6 The Evans Method for the Evans-Aronsson Problem; 6.7 An Energy Conservation Principle; 6.8 Porreta's Cross Estimates; 6.9 Bibliographical Notes; 7 A Priori Bounds for Stationary Models; 7.1 The Bernstein Method; 7.2 A MFG with Congestion; 7.3 Logarithmic Nonlinearity; 7.4 Bibliographical Notes
- 8 A Priori Bounds for Time-Dependent Models8.1 Subquadratic Hamiltonians; 8.2 Quadratic Hamiltonians; 8.3 Bibliographical Notes; 9 A Priori Bounds for Models with Singularities; 9.1 Logarithmic Nonlinearities; 9.2 Congestion Models: Local Existence; 9.2.1 Estimates for Arbitrary Terminal Time; 9.2.2 Short-Time Estimates; 9.3 Bibliographical Notes; 10 Non-local Mean-Field Games: Existence; 10.1 First-Order, Non-local Mean-Field Games; 10.2 Second-Order, Non-local Mean-Field Games; 10.3 Bibliographical Notes; 11 Local Mean-Field Games: Existence; 11.1 Bootstrapping Regularity
- Control code
- SPR958864856
- Dimensions
- unknown
- Extent
- 1 online resource.
- Form of item
- online
- Isbn
- 9783319389325
- Media category
- computer
- Media MARC source
- rdamedia
- Media type code
-
- c
- Specific material designation
- remote
- Label
- Regularity theory for mean field games systems, Diogo A. Gomes, Edgard A. Pimentel, Vardan Voskanyan
- Bibliography note
- Includes bibliographical references and index
- Carrier category
- online resource
- Carrier category code
-
- cr
- Carrier MARC source
- rdacarrier
- Content category
- text
- Content type code
-
- txt
- Content type MARC source
- rdacontent
- Contents
-
- Preface; Book Outline; Thanks; Bibliographical Notes; Acknowledgments; Contents; 1 Introduction; 1.1 Derivation of MFG Models; 1.1.1 Optimal Control and Hamilton-Jacobi Equations; 1.1.2 Transport Equation; 1.1.3 Mean-Field Models; 1.1.4 Extensions and Additional Problems; 1.1.5 Uniqueness; 2 Explicit Solutions, Special Transformations, and Further Examples; 2.1 Explicit Solutions; 2.2 The Hopf-Cole Transform; 2.3 Gaussian-Quadratic Solutions; 2.4 Interface Formation; 2.5 Bibliographical Notes; 3 Estimates for the Hamilton-Jacobi Equation; 3.1 Comparison Principle; 3.2 Control Theory Bounds
- 3.2.1 Optimal Trajectories3.2.2 Dynamic Programming Principle; 3.2.3 Subdifferentials and Superdifferentials of the Value Function; 3.2.4 Regularity of the Value Function; 3.3 Integral Bernstein Estimate; 3.4 Integral Estimates for HJ Equations; 3.5 Gagliardo-Nirenberg Estimates; 3.6 Bibliographical Notes; 4 Estimates for the Transport and Fokker-Planck Equations; 4.1 Mass Conservation and Positivity of Solutions; 4.2 Regularizing Effects of the Fokker-Planck Equation; 4.3 Fokker-Planck Equation with Singular Initial Conditions; 4.4 Iterative Estimates for the Fokker-Planck Equation
- 4.4.1 Regularity by Estimates on the Divergence of the Drift4.4.2 Polynomial Estimates for the Fokker-Planck Equation, p<∞; 4.4.3 Polynomial Estimates for the Fokker-Planck Equation, p=∞; 4.5 Relative Entropy; 4.6 Weak Solutions; 4.7 Bibliographical Notes; 5 The Nonlinear Adjoint Method; 5.1 Representation of Solutions and Lipschitz Bounds; 5.2 Conserved Quantities; 5.3 The Vanishing Viscosity Convergence Rate; 5.4 Semiconcavity Estimates; 5.5 Lipschitz Regularity for the Heat Equation; 5.6 Irregular Potentials; 5.7 The Hopf-Cole Transform; 5.8 Bibliographical Notes; 6 Estimates for MFGs
- 6.1 Maximum Principle Bounds6.2 First-Order Estimates; 6.3 Additional Estimates for Solutions of the Fokker-Plank Equation; 6.4 Second-Order Estimates; 6.4.1 Stationary Problems; 6.4.2 Time-Dependent Problems; 6.5 Some Consequences of Second-Order Estimates; 6.6 The Evans Method for the Evans-Aronsson Problem; 6.7 An Energy Conservation Principle; 6.8 Porreta's Cross Estimates; 6.9 Bibliographical Notes; 7 A Priori Bounds for Stationary Models; 7.1 The Bernstein Method; 7.2 A MFG with Congestion; 7.3 Logarithmic Nonlinearity; 7.4 Bibliographical Notes
- 8 A Priori Bounds for Time-Dependent Models8.1 Subquadratic Hamiltonians; 8.2 Quadratic Hamiltonians; 8.3 Bibliographical Notes; 9 A Priori Bounds for Models with Singularities; 9.1 Logarithmic Nonlinearities; 9.2 Congestion Models: Local Existence; 9.2.1 Estimates for Arbitrary Terminal Time; 9.2.2 Short-Time Estimates; 9.3 Bibliographical Notes; 10 Non-local Mean-Field Games: Existence; 10.1 First-Order, Non-local Mean-Field Games; 10.2 Second-Order, Non-local Mean-Field Games; 10.3 Bibliographical Notes; 11 Local Mean-Field Games: Existence; 11.1 Bootstrapping Regularity
- Control code
- SPR958864856
- Dimensions
- unknown
- Extent
- 1 online resource.
- Form of item
- online
- Isbn
- 9783319389325
- Media category
- computer
- Media MARC source
- rdamedia
- Media type code
-
- c
- Specific material designation
- remote
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