Stochastic Equations in Infinite Dimensions

Abstract

Now in its second edition, this book gives a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. In the first part the authors give a self-contained exposition of the basic properties of probability measure on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof. This revised edition includes two brand new chapters surveying recent developments in the area and an even more comprehensive bibliography, making this book an essential and up-to-date resource for all those working in stochastic differential equations.

Dynamical Behavior of Delayed Reaction–Diffusion Hopfield Neural Networks Driven by Infinite Dimensional Wiener Processes

“…The intensified noiseGd W represents response of the neuron to a current impulse associates with the local potential. According to [4]- [6],G is often assumed to have the formG(u) = u−u 0 , where u 0 is a constant.…”

“…But in a more realistic model, in order to describe the propagation of an electric potential in a neuron, it is sensible to include some noise in the system (see [4]). According to [18]- [20], noise disturbance is unavoidable in real nervous systems, even if the system is at rest, odd random impulse will arrive (see [4], [6]). Especially, the stochastic phenomenon often appears in the electrical circuit design of neural networks in application (see [16]).…”

A Model of Age–structured Population Under Stochastic Perturbation of Death and Birth Rates

“…The necessary condition of existence of the integrals in (3.1) is B 0 (u), B 1 (u) ∈ L 2 (H; H) (the space of Hilbert-Schmidt operators acting in H) for any u ∈ H. It is not the case here, therefore it is impossible to obtain theorems on existence and uniqueness of solution (weak, or mild) for the problem (3.3) (see, for example, Theorem 6.7, p. 164 in [5], Theorem 3.3, p. 97 in [6]). The way out can be found in setting the problem in the space of generalized Hilbert-spacevalued random variables (S) −ρ (H) ⊃ L 2 (Ω, F, P ; H), ρ ∈ [0; 1] (see the definition and properties of this space in [9]).…”

“…In section 3 we discuss difficulties that arise when we attempt to convert the difference equation into a stochastic differential equation in the Hilbert space H. We show that the use of the theory of Itô-type stochastic differential equations in infinite dimensional Hilbert spaces (see the review of the theory in [5,6]) is limited due to the properties of the operator coefficients in the difference equation obtained on the previous step. The necessary requirement for the operator-valued integrand of a well defined Itô integral with respect to a cylindrical Wiener process is the condition of being a Hilbert-Schmidt operator, which is not the case here.…”

Spectral Methods for Multiscale Stochastic Differential Equations

“…The proof of Theorem 4.7 relies on two results. First, it relies on the existence of a constant C L , proved in Appendix C, such that the effective drift and diffusion coefficients, obtained using (17) and (18) and by calculating A(x) as the unique symmetric positive semidefinite square root of D(x), satisfy…”