Stochastic partial differential equations with Lévy noise. An evolution equation approach.

*(English)*Zbl 1205.60122
Encyclopedia of Mathematics and Its Applications 113. Cambridge: Cambridge University Press (ISBN 978-0-521-87989-7/hbk). xii, 419 p. (2007).

The book provides an introduction to the theory of Stochastic Partial Differential Equations (SPDEs) with Lévy noise. It is divided into three parts: the first part is concerned with foundations, the second part treats existence and regularity of solutions and the third part discusses applications. The book concludes with several appendices providing some background material for ease of reference.

Part I consists of 8 chapters, the first one of which discusses the connections between stochastic dynamical systems, Markov processes and stochastic evolution equations with Lévy noise. In the second and third chapters analytical and probabilistic background material is collected, such as the definitions of Sobolev spaces, Lipschitz functions, Gaussian measures and semimartingales. Chapter 4 is devoted to the construction and properties of stochastic processes with stationary independent increments, i.e., Lévy processes in infinite-dimensional spaces, in particular in Hilbert and Banach spaces. It presents the two building blocks, the Poisson and the Wiener processes, as well as their properties, and the Lévy-Khinchin decomposition and the Lévy-Khinchin formula are proven. The short Chapter 5 treats Lévy semigroups and their generators. In Chapter 6 the concept of a Poisson random measure is introduced and its basic properties are established. Further, the latter are applied to a construction of Lévy processes, providing an additional interpretation of the Lévy-Khinchin formula. Chapter 7 presents the concept of a reproducing kernel Hilbert space for square integrable Lévy processes and the notion of cylindrical Lévy processes. The last chapter of the first part treats stochastic integration with respect to Hilbert-space-valued square-integrable martingales and, in particular, with respect to Lévy processes. A fundamental isometric formula is established and, moreover, integration with respect to a Poisson measure is introduced.

In the second part of the book the authors present results concerning existence, uniqueness and regularity of solutions to SPDEs driven by Lévy processes in a variety of settings. The first three chapters elaborate the general theory, which is then applied in Chapters 12 and 13, whereas Chapters 14 and 15 discuss additional cases. In more detail, Chapter 9 opens with the semigroup treatment of the abstract Cauchy problem for deterministic evolution problems. The authors then introduce the notions of weak and mild solutions for linear equations and prove their equivalence. Further, they prove the existence of weak solutions for general equations with Lipschitz-continuous coefficients and the Markov property. Chapter 10 treats existence and uniqueness of solutions to semilinear SPDEs with additive Lévy noise under the condition that both the linear and the nonlinear terms in the drift are of dissipative type. The stochastic reaction-diffusion equation is presented as a typical example. Chapter 11 is concerned with the regularity of solutions to SPDEs, in particular the time continuity of solutions to evolution equations driven by continuous processes. The main tool in this analysis is the so-called factorization method. In Chapters 12 and 13 the theory presented in the previous chapters is applied in a unified way to stochastic parabolic problems and to stochastic wave, delay and transport equations.

In Chapter 14 the authors introduce Lévy processes on the space of tempered distributions and compute their reproducing kernel spaces. This allows to discuss spatially homogeneous Lévy noise, e.g., impulsive cylindrical processes or coloured impulsive noise. The authors then establish results for semi-linear stochastic heat and wave equations driven by this type of noise. The last chapter of this part of the book treats the case of second order partial differential equations with Lévy noise only in the boundary conditions.

The third part of the book is devoted to special evolution equations and further results relevant in application problems. The first chapter of this part is concerned with existence and uniqueness of invariant measures for dissipative systems and provides an application of the results to the stochastic reaction-diffusion equation with a cubic nonlinearity and additive noise. Chapter 17 deals with evolution equations on a lattice and establishes results for the existence and uniqueness of solutions and the existence and exponential mixing of invariant measures. As an application problem in statistical mechanics the authors sketch the proof of the existence of a Gibbs measure for a particular class of spin systems. In Chapter 18 stochastic variants of a model for turbulence resulting in stochastic versions of the Burgers system, driven by a multiplicative cylindrical Wiener process or by additive Lévy noise, are studied and existence and uniqueness of mild and weak solutions established. In the short Chapter 19 it is shown how a model for environmental pollution can be described using stochastic parabolic equations driven by a Poisson measure. Chapter 20 is devoted to an application in finance. In particular, the authors propose and investigate a generalisation of the Heath, Jarrow and Morton model including Lévy noise.

Summarising, this book is an excellent addition to the literature on stochastic partial differential equations in general and in particular with respect to evolution equations driven by a discontinuous noise. The exposition is self-contained and very well written and, in my opinion, will become a standard tool for everyone working on stochastic evolution equations and related areas.

Part I consists of 8 chapters, the first one of which discusses the connections between stochastic dynamical systems, Markov processes and stochastic evolution equations with Lévy noise. In the second and third chapters analytical and probabilistic background material is collected, such as the definitions of Sobolev spaces, Lipschitz functions, Gaussian measures and semimartingales. Chapter 4 is devoted to the construction and properties of stochastic processes with stationary independent increments, i.e., Lévy processes in infinite-dimensional spaces, in particular in Hilbert and Banach spaces. It presents the two building blocks, the Poisson and the Wiener processes, as well as their properties, and the Lévy-Khinchin decomposition and the Lévy-Khinchin formula are proven. The short Chapter 5 treats Lévy semigroups and their generators. In Chapter 6 the concept of a Poisson random measure is introduced and its basic properties are established. Further, the latter are applied to a construction of Lévy processes, providing an additional interpretation of the Lévy-Khinchin formula. Chapter 7 presents the concept of a reproducing kernel Hilbert space for square integrable Lévy processes and the notion of cylindrical Lévy processes. The last chapter of the first part treats stochastic integration with respect to Hilbert-space-valued square-integrable martingales and, in particular, with respect to Lévy processes. A fundamental isometric formula is established and, moreover, integration with respect to a Poisson measure is introduced.

In the second part of the book the authors present results concerning existence, uniqueness and regularity of solutions to SPDEs driven by Lévy processes in a variety of settings. The first three chapters elaborate the general theory, which is then applied in Chapters 12 and 13, whereas Chapters 14 and 15 discuss additional cases. In more detail, Chapter 9 opens with the semigroup treatment of the abstract Cauchy problem for deterministic evolution problems. The authors then introduce the notions of weak and mild solutions for linear equations and prove their equivalence. Further, they prove the existence of weak solutions for general equations with Lipschitz-continuous coefficients and the Markov property. Chapter 10 treats existence and uniqueness of solutions to semilinear SPDEs with additive Lévy noise under the condition that both the linear and the nonlinear terms in the drift are of dissipative type. The stochastic reaction-diffusion equation is presented as a typical example. Chapter 11 is concerned with the regularity of solutions to SPDEs, in particular the time continuity of solutions to evolution equations driven by continuous processes. The main tool in this analysis is the so-called factorization method. In Chapters 12 and 13 the theory presented in the previous chapters is applied in a unified way to stochastic parabolic problems and to stochastic wave, delay and transport equations.

In Chapter 14 the authors introduce Lévy processes on the space of tempered distributions and compute their reproducing kernel spaces. This allows to discuss spatially homogeneous Lévy noise, e.g., impulsive cylindrical processes or coloured impulsive noise. The authors then establish results for semi-linear stochastic heat and wave equations driven by this type of noise. The last chapter of this part of the book treats the case of second order partial differential equations with Lévy noise only in the boundary conditions.

The third part of the book is devoted to special evolution equations and further results relevant in application problems. The first chapter of this part is concerned with existence and uniqueness of invariant measures for dissipative systems and provides an application of the results to the stochastic reaction-diffusion equation with a cubic nonlinearity and additive noise. Chapter 17 deals with evolution equations on a lattice and establishes results for the existence and uniqueness of solutions and the existence and exponential mixing of invariant measures. As an application problem in statistical mechanics the authors sketch the proof of the existence of a Gibbs measure for a particular class of spin systems. In Chapter 18 stochastic variants of a model for turbulence resulting in stochastic versions of the Burgers system, driven by a multiplicative cylindrical Wiener process or by additive Lévy noise, are studied and existence and uniqueness of mild and weak solutions established. In the short Chapter 19 it is shown how a model for environmental pollution can be described using stochastic parabolic equations driven by a Poisson measure. Chapter 20 is devoted to an application in finance. In particular, the authors propose and investigate a generalisation of the Heath, Jarrow and Morton model including Lévy noise.

Summarising, this book is an excellent addition to the literature on stochastic partial differential equations in general and in particular with respect to evolution equations driven by a discontinuous noise. The exposition is self-contained and very well written and, in my opinion, will become a standard tool for everyone working on stochastic evolution equations and related areas.

Reviewer: Evelyn Buckwar (Edinburgh)

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G51 | Processes with independent increments; Lévy processes |

35R60 | PDEs with randomness, stochastic partial differential equations |

34G99 | Differential equations in abstract spaces |