Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces
type: Item
instance of: scholarly article
title: Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces
zbMATH Open document ID: 1009.34074
author: Takeshi Taniguchi
author: Kai Liu
author: Aubrey Truman
published in: Journal of Differential Equations
publication date: 2002-07-08T00:00:00Z
review text: The authors consider the following abstract stochastic evolution equation in Hilbert space with time delay \[ dX(t)=(-AX(t)+f(t,X_t))dt+g(t,X_t)dW(t),\quad t\geq t_0, \] where \(-A\) is the generator of an analytic semigroup; \(X_t(u):=X(t+u)\) for \(u\in [-r,0]\) is the segment of the process from \(t-r\) to \(t\); \(f\) and \(g\) are some suitably regular functions and \(W\) is a Hilbert space-valued Wiener process with a nuclear covariance operator. Using semigroup theory, the authors give conditions for the existence and uniqueness of mild solutions to this equation and then find conditions for the \(p\)th moment and almost sure exponential stability of the solutions. Finally, the results are applied to a stochastic delay reaction-diffusion equation on a bounded interval with zero boundary conditions.
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1764709
zbMATH Keywords: mild solution
zbMATH Keywords: exponential stability
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: stochastic delay reaction-diffusion equation
reviewed by: Markus Reiß
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4073
OpenAlex ID: W2066118488
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DOI: 10.1006/JDEQ.2001.4073
description: scientific article; zbMATH DE number 1764709