Functional integro-differential stochastic evolution equations in Hilbert space
type: Item
instance of: scholarly article
title: Functional integro-differential stochastic evolution equations in Hilbert space
zbMATH Open document ID: 1031.60061
DOI: 10.1155/S1048953303000108
author: David N. Keck
author: Mark McKibben
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2003-11-10T00:00:00Z
full work available at URL: https://eudml.org/doc/50591
review text: Let \(K\) and \(H\) be real separable Hilbert spaces. Assume that \(W\) is a \(K\)-valued Wiener process with covariance operator \(Q\) and \(x_0\) is an \(H\)-valued random variable which is independent of \(W\). Consider the initial value problem of semilinear functional integro-differential stochastic evolution equations \[ x^\prime (t) = A x(t) + F(x)(t) + \int^t_0 G(x)(s) dW(s),\;0 \leq t \leq T, \quad x(0) = h(x) + x_0 \] with values in \(H\), where \(A : H \to H \) represents a linear operator, \(G : C([0,T],H) \to C([0,T], L^2(\Omega,BL(K,H)))\), \(F : C([0,T],H) \to L^p([0,T],L^2(\Omega,H))\) with \(1\leq p < + \infty\) and \(h: C([0,T],H) \to L^2_0(\Omega,H)\). The authors discuss global existence results concerning mild and periodic solutions under several growth and compactness conditions. Weak convergence of induced probability measures belonging to the family of finite-dimensional distributions of certain sequences of such stochastic equations is treated too. Basic proof-tools include Schaefer's fixed point theorem, techniques of linear semigroups and probability measures as well as results from infinite-dimensional SDEs. Conceivable applications to electromagnetic theory, population dynamics and heat conduction in materials with memory underline the importance of their work. An example of a nonlocal integro-partial SDE illustrates some thoughts of related abstract theory. Some necessary preliminaries compiled from probability theory and functional analysis ease the process of understanding by lesser experienced readership.
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 37H10
Mathematics Subject Classification ID: 37L55
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34K30
zbMATH DE Number: 2002245
zbMATH Keywords: semilinear stochastic evolution equations
zbMATH Keywords: mild solutions
zbMATH Keywords: semigroup techniques
zbMATH Keywords: Hilbert-space valued Wiener process
zbMATH Keywords: weak convergence of probability measures
zbMATH Keywords: stochastic equations with memory
zbMATH Keywords: Schaefer's fixed point theorem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2002245