Differential Equations on Measures and Functional Spaces (Birkhäuser Advanced Texts Basler Lehrbücher)

Contents
Preface
	Objectives, scope and methodology
	General context and specific features
	Readers and prerequisites
	Bibliographic comments
	Acknowledgement
Standard notations
	Standard abbreviations
Chapter 1 Analysis on Measures and Functional Spaces
	1.1 Banach spaces: notations and examples
	1.2 Smooth functions on Banach spaces
	1.3 Additive and multiplicative integrals
	1.4 Differentials of the norms
	1.5 Smooth mappings between Banach spaces
	1.6 Locally convex spaces and Fréchet spaces
	1.7 Linear operators in spaces of measures and functions
	1.8 Fractional calculus
	1.9 Generalized functions: main operations
	1.10 Generalized functions: regularization
	1.11 Fourier transform, fundamental solutions and Green functions
	1.12 Sobolev spaces
	1.13 Variational derivatives
	1.14 Derivatives compatible with duality, AMand AL-spaces
	1.15 Hints and answers to chosen exercises
	1.16 Summary and comments
Chapter 2 Basic ODEs in Complete Locally Convex Spaces
	2.1 Fixed-point principles for curves in Banach spaces
	2.2 ODEs in Banach spaces: well-posedness
	2.3 Linear equations and chronological exponentials
	2.4 Linear evolutions involving spatially homogeneous ΨDOs
	2.5 Hamiltonian systems, boundary-value problems and the method of shooting
	2.6 Hamilton–Jacobi equation, method of characteristics and calculus of variation
	2.7 Hamilton–Jacobi–Bellman equation and optimal control
	2.8 Sensitivity of integral equations
	2.9 ODEs in Banach spaces: sensitivity
	2.10 Linear first-order partial differential equations
	2.11 Equations with memory: causality
	2.12 Equations with memory: fractional derivatives
	2.13 Linear fractional ODEs and related integral equations
	2.14 Linear fractional evolutions involving spatially homogeneous ΨDOs
	2.15 Sensitivity of integral and differential equations: advanced version
	2.16 ODEs in locally convex spaces
	2.17 Monotone and accretive operators
	2.18 Hints and answers to chosen exercises
	2.19 Summary and comments
Chapter 3 Discrete Kinetic Systems: Equations in l+p
	3.1 Equations in Rn+
	3.2 Examples in Rn+: replicator dynamics and mass-action-law kinetics
	3.3 Entropy and equilibria for linear evolutions in Rn+
	3.4 Entropy and equilibria for nonlinear evolutions in Rn+
	3.5 Kinetic equations for collisions, fragmentation, reproduction and preferential attachment
	3.6 Simplest equations in l+p
	3.7 Existence of solutions for equations in l+p
	3.8 Additive bounds for rates
	3.9 Evolution of moments under additive bounds
	3.10 Accretive operators in lp
	3.11 Accretivity for evolutions with additive rates
	3.12 The major well-posedness result in l+p
	3.13 Sensitivity
	3.14 Second-order sensitivity
	3.15 Stability of solutions with respect to coefficients
	3.16 Hints and answers to chosen exercises
	3.17 Summary and comments
Chapter 4 Linear Evolutionary Equations: Foundations
	4.1 Semigroups and their generators
	4.2 Semigroups of operators on Banach spaces
	4.3 Simple diffusions and the Schrödinger equation
	4.4 Evolutions generated by powers of the Laplacian
	4.5 Evolutions generated by ΨDOs with homogeneous symbols and their mixtures
	4.6 Perturbation theory and the interaction picture
	4.7 Path integral representation
	4.8 Diffusion with drifts and Schrödinger equations with singular potentials and magnetic fields
	4.9 Propagators and their generators
	4.10 Well-posedness of linear Cauchy problems
	4.11 The operator-valued Riccati equation
	4.12 An infinite-dimensional diffusion equation in variational derivatives
	4.13 Perturbation theory for propagators
	4.14 Diffusions and Schrödinger equations with nonlocal terms
	4.15 ΨDOs with homogeneous symbols (time-dependent case)
	4.16 Higher-order ΨDEs with nonlocal terms
	4.17 Hints and answers to chosen exercises
	4.18 Summary and comments
Chapter 5 Linear Evolutionary Equations: Advanced Theory
	5.1 T-products with three-level Banach towers
	5.2 Adding generators with 4-level Banach towers
	5.3 Mixing generators
	5.4 The method of frozen coefficients: heuristics
	5.5 The method of frozen coefficients: estimates for the Green function
	5.6 The method of frozen coefficients: main examples
	5.7 The method of frozen coefficients: regularity
	5.8 The method of frozen coefficients: the Cauchy problem
	5.9 Uniqueness via duality and accretivity; generalized solutions
	5.10 Uniqueness via positivity and approximations; Feller semigroups
	5.11 Lévy–Khintchin generators and convolution semigroups
	5.12 Potential measures
	5.13 Vector-valued convolution semigroups
	5.14 Equations of order at most one
	5.15 Smoothness and smoothing of propagators
	5.16 Summary and comments
Chapter 6 The Method of Propagators for Nonlinear Equations
	6.1 Hamilton–Jacobi–Bellman (HJB) and Ginzburg–Landau equations
	6.2 Higher-order PDEs and ΨDEs, and Cahn–Hilliard-type equations
	6.3 Nonlinear evolutions and multiplicative-integral equations
	6.4 Causal equations and general path-dependent equations
	6.5 Simplest nonlinear diffusions: weak treatment
	6.6 Simplest nonlinear diffusions: strong treatment
	6.7 Simplest nonlinear diffusions: regularity and sensitivity
	6.8 McKean–Vlasov equations
	6.9 Landau–Fokker–Planck-type equations
	6.10 Forward-backward systems
	6.11 Linearized evolution around non-linear propagators
	6.12 Sensitivity of nonlinear propagators
	6.13 Summary and comments
Chapter 7 Equations in Spaces of Weighted Measures
	7.1 Conditional positivity
	7.2 Simplest equations that preserve positivity
	7.3 Path-dependent equations and forward-backward systems
	7.4 Kinetic equations (Boltzmann, Smoluchowski, Vlasov, Landau) and replicator dynamics
	7.5 Well-posedness for basic kinetic equations
	7.6 Equations with additive bounds for rates
	7.7 On the sensitivity of kinetic equations
	7.8 On the derivation of kinetic equations: second quantization and beyond
	7.9 Interacting particles and measure-valued diffusions
	7.10 Summary and comments
Chapter 8 Generalized Fractional Differential Equations
	8.1 Green functions of fractional derivatives and the Mittag-Leffler function
	8.2 Linear evolution
	8.3 The fractional HJB equation and related equations with smoothing generators
	8.4 Generalized fractional integration and differentiation
	8.5 Generalized fractional linear equations, part I
	8.6 Generalized fractional linear equations, part II
	8.7 The time-dependent case; path integral representation
	8.8 Chronological operator-valued Feynmann–Kac formula
	8.9 Summary and comments
Chapter 9 Appendix
	9.1 Fixed-point principles
	9.2 Special functions
	9.3 Asymptotics of the Fourier transform: power functions and their exponents
	9.4 Asymptotics of the Fourier transform: functions of power growth
	9.5 Argmax in convex Hamiltonians
Bibliography
Index