Stochastic Partial Differential Equations with Levy Noise

Transcription

1 Stochastic Partial Differential Equations with Levy Noise An Evolution Equation Approach S..PESZAT and J. ZABCZYK Institute of Mathematics, Polish Academy of Sciences' CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface page ix Parti Foundations 1 1 Why equations with Levy noise? Discrete-time dynamical systems Deterministic continuous-time systems ' ' ' Stochastic continuous-time systems Courrege's theorem Ito's approach Infinite-dimensional case 12 2 Analytic preliminaries Notation ' Sobolev and Holder spaces L p - and C p -spaces " ' ' Lipschitz functions and composition operators Differential operators 17 3 Probabilistic preliminaries 20 "3.1 Basic definitions ~ Kolmogorov existence theorem Random elements in Banacti spaces Stochastic processes in Banach spaces Gaussian measures on Hilbert spaces Gaussian 1 measures on topological spaces 30 ' 3.7 Submartingales, ; Semimartingales Burkholder-Davies-Guhdy inequalities. 37

3 vi Contents 4 Levy processes Basic properties Two building blocks - Poisson and Wiener processes Compound Poisson processes in a Hilbert space Wiener processes in a Hilbert space Levy-Khinchin decomposition Levy-Khinchin formula Laplace transforms of convolution semigroups Expansion with respect to an orthonormal basis Square integrable Levy processes Levy processes on Banach spaces 72 5 Levy semigroups Basic properties ' Generators 78 6 Poisson random measures Introduction Stochastic integral of deterministic fields Application to construction of Levy processes Moment estimates in Banach spaces 90 7 Cylindrical processes and reproducing kernels Reproducing kernel Hilbert space Cylindrical Poisson processes _, Compensated Poisson measure as a martingale Stochastic integration Operator-valued angle bracket process ; Construction of the stochastic integral Space of integrands Local properties of stochastic integrals, Stochastic Fubini theorem ', Stochastic integral with respect to a Levy process, Integration with respect to a Poisson random measure L p -theory for vector-valued integrands 130 Part II Existence and Regularity General existence and,uniqueness results Deterministic linear equations ' Mild solutions <\ Equivalence of weak and mild solutions Linear equations 155

4 Contents vii 9.5 Existence of weak solutions Markov property x Equations with general Levy processes Generators and a martingale problem Equations with non-lipschitz coefficients Dissipative mappings Existence theorem Reaction-diffusion equation Factorization and regularity Finite-dimensional case Infinite-dimensional case Applications to time continuity The case of an arbitrary martingale Stochastic parabolic problems Introduction Space-time continuity in the Wiener case The jump case Stochastic heat equation Equations with fractional Laplacian and stable noise Wave and delay equations Stochastic wave equation on [0, 1] Stochastic wave equation on M. d driven by impulsive noise Stochastic delay equations Equations driven by a spatially homogeneous noise Tempered distributions Levy processes in S'(R d ) RKHS of a square integrable Levy process in S'(R d ) Spatially homogeneous Levy processes Examples RKHS of a homogeneous noise Stochastic equations on R d Stochastic heat equation Space-time regularity in the Wiener case Stochastic wave equation Equations with noise on the boundary Introduction Weak and mild solutions Analytical preliminaries L 2 case Poisson perturbation 282

5 viii Contents Part III Applications Invariant measures Basic definitions Existence results Invariant measures for the reaction-diffusion equation Lattice systems Introduction Global interactions Regular case Non-Lipschitz case Kolmogorov's formula Gibbs measures Stochastic Burgers equation Burgers system Uniqueness and local existence of solutions Stochastic Burgers equation with additive noise Environmental pollution model Model Bond market models Forward curves and the HJM postulate ' ' HJM condition HJMM equation Linear volatility. ^ 20.5 BGM equation Consistency problem, 350 Appendix A Operators oh Hilbert spaces 355 Appendix B Co-semigroups 365 Appendix C Regularization of Markov processes 388 Appendix D ltd formulae 391 Appendix E Levy-Khinchin formula on [0, +oo) 394 Appendix F Proof of Lemma List of symbols 399 References 403 Index 415