Stochastic Partial Differential Equations with LГ©vy Noise: An Evolution Equation Approach

Cover
Title page
Preface
Part I Foundations
1 Why equations with Lévy noise?
	1.1 Discrete-time dynamical systems
	1.2 Deterministic continuous-time systems
	1.3 Stochastic continuous-time systems
	1.4 Courrège\'s theorem
	1.5 Itô\'s approach
	1.6 Infinite-dimensional case
2 Analytic preliminaries
	2.1 Notation
	2.2 Sobolev and Hölder spaces
	2.3 L^p and C_ρ-spaces
	2.4 Lipschitz functions and composition operators
	2.5 Differential operators
3 Probabilistic preliminaries
	3.1 Basic definitions
	3.2 Kolmogorov existence theorem
	3.3 Random elements in Banach spaces
	3.4 Stochastic processes in Banach spaces
	3.5 Gaussian measures on Hilbert spaces
	3.6 Gaussian measures on topological spaces
	3.7 Submartingales
	3.8 Semimartingales
	3.9 Burkholder-Davies-Gundy inequalities
4 Lévy processes
	4.1 Basic properties
	4.2 Two building blocks - Poisson and Wiener processes
	4.3 Compound Poisson processes in a Hilbert space
	4.4 Wiener processes in a Hilbert space
	4.5 Lévy-Khinchin decomposition
	4.6 Lévy-Khinchin formula
	4.7 Laplace transforms of convolution semigroups
	4.8 Expansion with respect to an orthonormal basis
	4.9 Square integrable Lévy processes
	4.10 Lévy processes on Banach spaces
5 Lévy semigroups
	5.1 Basic properties
	5.2 Generators
6 Poisson random measures
	6.1 Introduction
	6.2 Stochastic integral of deterministic fields
	6.3 Application to construction of Lévy processes
	6.4 Moment estimates in Banach spaces
7 Cylindrical processes and reproducing kernels
	7.1 Reproducing kernel Hilbert space
	7.2 Cylindrical Poisson processes
	7.3 Compensated Poisson measure as a martingale
8 Stochastic integration
	8.1 Operator-valued angle bracket process
	8.2 Construction of the stochastic integral
	8.3 Space of integrands
	8.4 Local properties of stochastic integrals
	8.5 Stochastic Fubini theorem
	8.6 Stochastic integral with respect to a Lévy process
	8.7 Integration with respect to a Poisson random measure
	8.8 L^p-theory for vector-valued integrands
Part II Existence and Regularity
9 General existence and uniqueness results
	9.1 Deterministic linear equations
	9.2 Mild solutions
	9.3 Equivalence of weak and mild solutions
	9.4 Linear equations
	9.5 Existence of weak solutions
	9.6 Markov property
	9.7 Equations with general Lévy processes
	9.8 Generators and a martingale problem
10 Equations with non-Lipschitz coefficients
	10.1 Dissipative mappings
	10.2 Existence theorem
	10.3 Reaction-diffusion equation
11 Factorization and regularity
	11.1 Finite-dimensional case
	11.2 Infinite-dimensiona1 case
	11.3 Applications to time continuity
	11.4 The case of an arbitrary martingale
12 Stochastic parabolic problems
	12.1 Introduction
	12.2 Space-time continuity in the Wiener case
	12.3 The jump case
	12.4 Stochastic heat equation
	12.5 Equations with fractional Laplacian and stable noise
13 Wave and delay equations
	13.1 Stochastic wave equation on [0,1]
	13.2 Stochastic wave equation on R^d driven by impulsive noise
	13.3 Stochastic delay equations
14 Equations driven by a spatially homogeneous noise
	14.1 Tempered distributions
	14.2 Lévy processes in S\'(R^d)
	14.3 RKHS of a square integrable Lévy process in S\'(R^d)
	14.4 Spatially homogeneous Lévy processes
	14.5 Examples
	14.6 RKHS of a homogeneous noise
	14.7 Stochastic equations on R^d
	14.8 Stochastic heat equation
	14.9 Space-time regularity in the Wiener case
	14.10 Stochastic wave equation
15 Equations with noise on the boundary
	15.1 Introduction
	15.2 Weak and mild solutions
	15.3 Analytical preliminaries
	15.4 L² case
	15.5 Poisson perturbation
Part III Applications
16 Invariant measures
	16.1 Basic definitions
	16.2 Existence results
	16.3 Invariant measures for the reaction-diffusion equation
17 Lattice systems
	17.1 Introduction
	17.2 Global interactions
	17.3 Regular case
	17.4 Non-Lipschitz case
	17.5 Kolmogorov\'s formula
	17.6 Gibbs measures
18 Stocbastic Burgers equation
	18.1 Burgers system
	18.2 Uniqueness and local existence of solutions
	18.3 Stochastic Burgers equation with additive noise
19 Environmental pollution model
	19.1 Model
20 Bond market models
	20.1 Forward curves and the HJM postulate
	20.2 HJM condition
	20.3 HJMM equation
	20.4 Linear volatility
	20.5 BGM equation
	20.6 Consistency problem
Appendix A Operators on Hilbert spaces
Appendix B Co-semigroups
Appendix C Regularization of Markov processes
Appendix D Itô formulae
Appendix E Lévy-Khinchin formula on [0,+∞)
Appendix F Proof of Lemma 4.24
List of symbols
References
Index