The book is devoted to stochastic evolution equations with Lévy noise. Such equations are important because, roughly speaking, stochastic dynamical systems, or equivalently Markov processes, can be represented as solutions to such equations. In this introductory chapter, it is shown how that is the case. To motivate better the construction of the associated stochastic equations, the chapter starts with discrete-time systems.

Discrete-time dynamical systems

A deterministic discrete-time dynamical system consists of a set E, usually equipped with a σ-field ε of subsets of itself, and a mapping F, usually measurable, acting from E into E. If the position of the system at time t = 0, 1, …, is denoted by X(t) then by definition X(t + 1) = F(X(t)), t = 0, 1, … The sequences (X(t), t = 0, 1, …) are the so-called trajectories or paths of the dynamical system, and their asymptotic properties are of prime interest in the theory. The set E is called the state space and the transformation F determines the dynamics of the system.

If the present state x determines only the probability P(x, Γ) that at the next moment the system will be in the set Γ then one says that the system is stochastic. Thus a stochastic dynamical system consists of the state space E, a σ-field ε and a function P = P(x, Γ), x ∈ E, Γ ∈ ε, such that, for each Γ ∈ ε, P(·, Γ) is a measurable function and, for each x ∈ E, P(x, ·) is a probability measure. We call P the transition function or transition probability.

This book is an introduction to the theory of stochastic evolution equations with Lévy noise. The theory extends several results known for stochastic partial differential equations (SPDEs) driven by Wiener processes. We develop a general framework and discuss several classes of examples both with general Lévy noise and with Wiener noise. Our approach is functional analytic and, as in Da Prato and Zabczyk (1992a), SPDEs are treated as ordinary differential equations in infinitedimensional spaces with irregular coefficients. In many respects the Lévy noise theory is similar to that for Wiener noise, especially when the driving Lévy process is a square integrable martingale. The general case reduces to this, owing to the Lévy–Khinchin decomposition. The functional analytic approach also allows us to treat equations with a so-called cylindrical Lévy noise and implies, almost automatically, that solutions to equations with Lévy noise are Markovian. In some important cases, however, a càdlàg version of the solution does not exist.

An important role in our approach is played by a generalization of the concept of the reproducing kernel Hilbert space to non-Gaussian random variables, and its independence of the space in which the random variable takes values. In some cases it proves useful to treat Poissonian random measures, with respect to which many SPDEs have been studied, as Lévy processes properly defined in appropriate state spaces.

The majority of the results appear here for the first time in book form, and the book presents several completely new results not published previously, in particular, for equations driven by homogeneous noise and for dissipative systems.