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 meanfield games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for wellposedness in the context of meanfield problems, including stationary and timedependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of meanfield couplings. It also explores stationary and timedependent MFGs through a series of apriori estimates for solutions of the HamiltonJacobi and FokkerPlanck equation. It shows sophisticated apriori 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 HamiltonJacobi Equations; 1.1.2 Transport Equation; 1.1.3 MeanField 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 HopfCole Transform; 2.3 GaussianQuadratic Solutions; 2.4 Interface Formation; 2.5 Bibliographical Notes; 3 Estimates for the HamiltonJacobi 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 GagliardoNirenberg Estimates; 3.6 Bibliographical Notes; 4 Estimates for the Transport and FokkerPlanck Equations; 4.1 Mass Conservation and Positivity of Solutions; 4.2 Regularizing Effects of the FokkerPlanck Equation; 4.3 FokkerPlanck Equation with Singular Initial Conditions; 4.4 Iterative Estimates for the FokkerPlanck Equation
 4.4.1 Regularity by Estimates on the Divergence of the Drift4.4.2 Polynomial Estimates for the FokkerPlanck Equation, p<∞; 4.4.3 Polynomial Estimates for the FokkerPlanck 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 HopfCole Transform; 5.8 Bibliographical Notes; 6 Estimates for MFGs
 6.1 Maximum Principle Bounds6.2 FirstOrder Estimates; 6.3 Additional Estimates for Solutions of the FokkerPlank Equation; 6.4 SecondOrder Estimates; 6.4.1 Stationary Problems; 6.4.2 TimeDependent Problems; 6.5 Some Consequences of SecondOrder Estimates; 6.6 The Evans Method for the EvansAronsson 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 TimeDependent 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 ShortTime Estimates; 9.3 Bibliographical Notes; 10 Nonlocal MeanField Games: Existence; 10.1 FirstOrder, Nonlocal MeanField Games; 10.2 SecondOrder, Nonlocal MeanField Games; 10.3 Bibliographical Notes; 11 Local MeanField 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 meanfield games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for wellposedness in the context of meanfield problems, including stationary and timedependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of meanfield couplings. It also explores stationary and timedependent MFGs through a series of apriori estimates for solutions of the HamiltonJacobi and FokkerPlanck equation. It shows sophisticated apriori 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 HamiltonJacobi Equations; 1.1.2 Transport Equation; 1.1.3 MeanField 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 HopfCole Transform; 2.3 GaussianQuadratic Solutions; 2.4 Interface Formation; 2.5 Bibliographical Notes; 3 Estimates for the HamiltonJacobi 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 GagliardoNirenberg Estimates; 3.6 Bibliographical Notes; 4 Estimates for the Transport and FokkerPlanck Equations; 4.1 Mass Conservation and Positivity of Solutions; 4.2 Regularizing Effects of the FokkerPlanck Equation; 4.3 FokkerPlanck Equation with Singular Initial Conditions; 4.4 Iterative Estimates for the FokkerPlanck Equation
 4.4.1 Regularity by Estimates on the Divergence of the Drift4.4.2 Polynomial Estimates for the FokkerPlanck Equation, p<∞; 4.4.3 Polynomial Estimates for the FokkerPlanck 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 HopfCole Transform; 5.8 Bibliographical Notes; 6 Estimates for MFGs
 6.1 Maximum Principle Bounds6.2 FirstOrder Estimates; 6.3 Additional Estimates for Solutions of the FokkerPlank Equation; 6.4 SecondOrder Estimates; 6.4.1 Stationary Problems; 6.4.2 TimeDependent Problems; 6.5 Some Consequences of SecondOrder Estimates; 6.6 The Evans Method for the EvansAronsson 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 TimeDependent 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 ShortTime Estimates; 9.3 Bibliographical Notes; 10 Nonlocal MeanField Games: Existence; 10.1 FirstOrder, Nonlocal MeanField Games; 10.2 SecondOrder, Nonlocal MeanField Games; 10.3 Bibliographical Notes; 11 Local MeanField 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 HamiltonJacobi Equations; 1.1.2 Transport Equation; 1.1.3 MeanField 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 HopfCole Transform; 2.3 GaussianQuadratic Solutions; 2.4 Interface Formation; 2.5 Bibliographical Notes; 3 Estimates for the HamiltonJacobi 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 GagliardoNirenberg Estimates; 3.6 Bibliographical Notes; 4 Estimates for the Transport and FokkerPlanck Equations; 4.1 Mass Conservation and Positivity of Solutions; 4.2 Regularizing Effects of the FokkerPlanck Equation; 4.3 FokkerPlanck Equation with Singular Initial Conditions; 4.4 Iterative Estimates for the FokkerPlanck Equation
 4.4.1 Regularity by Estimates on the Divergence of the Drift4.4.2 Polynomial Estimates for the FokkerPlanck Equation, p<∞; 4.4.3 Polynomial Estimates for the FokkerPlanck 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 HopfCole Transform; 5.8 Bibliographical Notes; 6 Estimates for MFGs
 6.1 Maximum Principle Bounds6.2 FirstOrder Estimates; 6.3 Additional Estimates for Solutions of the FokkerPlank Equation; 6.4 SecondOrder Estimates; 6.4.1 Stationary Problems; 6.4.2 TimeDependent Problems; 6.5 Some Consequences of SecondOrder Estimates; 6.6 The Evans Method for the EvansAronsson 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 TimeDependent 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 ShortTime Estimates; 9.3 Bibliographical Notes; 10 Nonlocal MeanField Games: Existence; 10.1 FirstOrder, Nonlocal MeanField Games; 10.2 SecondOrder, Nonlocal MeanField Games; 10.3 Bibliographical Notes; 11 Local MeanField 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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