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## Über dieses Buch

This book lays the foundations for a theory on almost periodic stochastic processes and their applications to various stochastic differential equations, functional differential equations with delay, partial differential equations, and difference equations. It is in part a sequel of authors recent work on almost periodic stochastic difference and differential equations and has the particularity to be the first book that is entirely devoted to almost periodic random processes and their applications. The topics treated in it range from existence, uniqueness, and stability of solutions for abstract stochastic difference and differential equations.

## Inhaltsverzeichnis

### Chapter 1. Banach and Hilbert Spaces

Banach and Hilbert spaces play a key role in functional analysis. This chapter provides the reader with a detailed account of properties of Banach and Hilbert spaces. Various examples of Banach and Hilbert spaces are also discussed, among them are quotient spaces, $$L^p(\mathcal{O})$$ spaces, $$BC(\mathbb{R}, \mathcal{B})$$ spaces, Hölder spaces $$C^{\alpha}(J, \mathcal{B})$$, and Sobolev spaces $$W^{k, p}(\mathcal{O})$$.
Paul H. Bezandry, Toka Diagana

### Chapter 2. Bounded and Unbounded Linear Operators

This chapter is devoted to the basic material on operator theory, semigroups, evolution familites, interpolation spaces, fractional powers of operators, intermediate spaces, and their basic properties needed in the sequel. Various illustrative examples are discussed in-depth. The estimates in Lemma 2.2 (Diagana et al. [52]) and Lemma 2.4 (Diagana [62]) will play a key role throughout the book. Detailed proofs of these lemmas are discussed.
Paul H. Bezandry, Toka Diagana

### Chapter 3. An Introduction to Stochastic Differential Equations

Chapter 3 develops probabilistic tools needed for the analysis of stochastic problems in the book. It begins with the review of the fundamentals of probability including the notion of conditional expectation, which is very useful in the sequel. This chapter also offers an introduction to the mathematical theory of stochastic processes, including the notion of continuity, measurability, stopping times, martingales,Wiener processes, and Gaussian processes. These concepts enable us to define the so-called Itô integral, the Itô formula, and diffusion processes. An extension of Itô integrals to Hilbert spaces and stochastic convolution integrals are also discussed. An investigation on stochastic differential equations driven by Wiener processes is given at end of the chapter. Special emphasis will be on the boundedness and stability of solutions.
Paul H. Bezandry, Toka Diagana

### Chapter 4. P-th Mean Almost Periodic Random Functions

Chapter 4 introduces the concept of p-th mean almost periodicity is introduced. It is shown that each p-th mean almost periodic stochastic process defined on a probability space ($$\Omega, \mathcal{F}, \mathbf{P}$$) is uniformly continuous and stochastically bounded. The collection of such stochastic processes is a Banach space when it is equipped with its natural norm. Moreover, two composition theorems for p-th mean almost periodic processes (Theorem 4.4 and Theorem 4.5) are established. They play a crucial role in the study of the existence (and uniqueness) of p-th mean almost periodic solutions to various stochastic differential equations on $$L^{P} (\Omega, \mathbb{H})$$ where $$\mathbb{H}$$ is a real separable Hilbert space.
Paul H. Bezandry, Toka Diagana

### Chapter 5. Existence Results for Some Stochastic Differential Equations

Chapter 5 is devoted to the study of the existence of p-th mean almost periodic solutions to some classes of nonautonomous stochastic differential equations of type
$$dX(t)=A(t)X(t)dt + F(t, X(t))dt + G(t, X(t))d\mathbb{W}(t), \ \ t \in \mathbb{R},$$
where $$(A(t))_{t \in \mathbb{R}}$$ is a family of densely defined closed linear operators satisfying the wellknown Acquistapace-Terreni conditions, $$F : \mathbb{R} \times L^p (\Omega, \mathbb{H}) \to L^p (\Omega, \mathbb{H}) \ {\rm and} \ G : \mathbb{R} \times L^p (\Omega, \mathbb{H}) \to L^p(\Omega, \mathbb{L}^{0}_{2})$$ are jointly continuous satisfying some additional conditions, and $$\mathbb{W}$$ is a Q-Wiener process with values in $$\mathbb{K}$$. Some sufficient conditions for the existence of p-th mean almost periodic solutions to the autonomous counterpart of the above equation are also obtained. Finally, an analysis of some N-dimensional parabolic stochastic partial differential equations is provided to illustrate the applicability of our abstract results.
Paul H. Bezandry, Toka Diagana

### Chapter 6. Existence Results for Some Partial Stochastic Differential Equations

Chapter 6 deals with the existence of p-th mean almost periodic mild solutions for some classes of stochastic partial evolution equations with infinite delay of type
$$\begin{array}{lll}d[X(\omega, t) + f_1(t, X_t, (\omega))] & = & \left[\mathcal{A}X(\omega, t) + f_2(t, X_t(\omega))\right]dt \\ {} & + & f_3(t, X_t(\omega))d \mathbb{W}(\omega, t), \ t \in \mathbb{R}, \ \omega \in \Omega ,\end{array}$$
where $$\mathcal{A} : \mathcal{D} = \mathcal{D}(\mathcal{A})\subset \mathbb{H} \to \mathbb{H}$$ is a sectorial linear operator whose corresponding analytic semigroup is hyperbolic, that is, $$\sigma (\mathcal{A})\cap i\mathbb{R}$$ is empty, and $$f_1 : \mathbb{R} \times \mathbb{H} \to \mathbb{H}_{\beta} (0< \alpha < \frac{1}{p} < \beta < 1), \ f_2 : \mathbb{R} \times \mathbb{H} \to \mathbb{H}, \ {\rm and} \ f_3 : \mathbb{R} \times \mathbb{H} \to \mathbb{L}^{0}_{2}$$ are jointly continuous functions. Chapter 6 also presents some recent results on the existence of p-th mean almost periodic and Stepanov almost periodic (mild) solutions to various nonautonomous differential equations using the well-known Schauder fixed point theorem. Some examples are also discussed.
Paul H. Bezandry, Toka Diagana

### Chapter 7. Existence Results For Some Second-Order Stochastic Differential Equations

The impetus of the work done in this chapter comes from two main sources from the deterministic setting. The first one is the work of Mawhin [139], in which the dissipativeness and the existence of bounded solutions on the whole real number line to the second-order differential equations given by
$$u^{\prime \prime}(t) + cu^{\prime} + Au + g(t, u) = 0, \ \ t \in \mathbb{R},$$
where $$A : D(A) \subset \mathbb{H} \to \mathbb{H}$$ is a self-adjoint operator on a Hilbert space $$\mathbb{H}$$, which is semipositive definite and has a compact resolvent,$$c>0, \ {\rm and} \ g : \mathbb{R} \times \mathbb{H} \to \mathbb{H}$$ is bounded, sufficiently regular, and satisfies some semi-coercivity condition, was established. The abstract results in [139] were subsequently utilized to study the existence of bounded solutions to the so-called nonlinear telegraph equation subject to some Neumann boundary conditions. Unfortunately, the main result of this chapter does not apply to the telegraph equation as the linear operator presented in [139], which involves Neumann boundary boundary conditions, lacks exponential dichotomy.
The second source is the work by Leiva [118], in which the existence of (exponentially stable) bounded solutions and almost periodic solutions to the second-order systems of differential equations given by
$$u^{\prime\prime}(t) + cu^{\prime}(t) + dAu +kH(u)=P(t),\quad u\in \mathbb{R}^n, \quad t\in \mathbb{R},$$
where $$A \ {\rm is \ an} \ n \times n$$-matrix whose eigenvalues are positive, c, d, k are positive constants, $$H : \mathbb{R}^n \to \mathbb{R}^n$$ is a locally Lipschitz function, $$P : \mathbb{R} \to \mathbb{R}^n$$ is a bounded continuous function, was established.
In this chapter, using slightly different techniques as in [118, 139], we study and obtain some reasonable sufficient conditions, which do guarantee the existence of square-mean almost periodic solutions to the classes of nonautonomous second-order stochastic differential equations
$$\begin{array}{lll}dX^{\prime}(\omega, t) + a(t) dX(\omega, t) & = & \left[ -b(t) \mathcal{A}X(\omega, t) + f_1(t, X(\omega, t))\right]dt \\ {} & {} & +f_2(t, X(\omega, t)) d\mathbb{W}(\omega, t), \end{array}$$
for all $$\omega \in \Omega \ {\rm and} \ t\in \mathbb{R}, \ {\rm where} \ \mathcal{A} : D(\mathcal{A}) \subset \mathbb{H} \to \mathbb{H}$$ is a self-adjoint linear operator whose spectrum consists of isolated eigenvalues $$0 < \lambda_1 < \lambda_2 < \ldots < \lambda_n \to \infty$$ with each eigenvalue having a finite multiplicity $$\gamma_j$$ equals to the multiplicity of the corresponding eigenspace, the functions $$a, b : \mathbb{R} \to (0, \infty)$$ are almost periodic functions, and the function $$f_i(i = 1, 2) : \mathbb{R} \times L^2(\Omega, \mathbb{H}) \to L^2(\Omega, \mathbb{H})$$ are jointly continuous functions satisfying some additional conditions and $$\mathbb{W}$$ is a one dimensional Brownian motion.
Paul H. Bezandry, Toka Diagana

### Chapter 8. Mean Almost Periodic Solutions to Some Stochastic Difference Equations

Periodicity often appears in implicit ways in various natural phenomena. For instance, this is the case when one studies the effects of fluctuating environments on population dynamics. Though one can deliberately periodically fluctuate environmental parameters in controlled laboratory experiments, fluctuations in nature are hardly periodic. Almost periodicity is more likely to accurately describe natural fluctuations (Diagana et al. [63]). This chapter deals with discrete-time stochastic processes known as random sequences. Here, we are particularly interested in the study of almost periodicity of those random sequences and their applications to stochastic difference equations.
Paul H. Bezandry, Toka Diagana

### Backmatter

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