Luo, Jiaowan Stability of stochastic partial differential equations with infinite delays. (English) Zbl 1151.60336 J. Comput. Appl. Math. 222, No. 2, 364-371 (2008). Summary: In this paper, we study the existence and the asymptotical stability in \(p\)-th moment of mild solutions to stochastic partial differential equations with infinite delays \[ \begin{cases} dx(t)=[Ax(t)+f(t,x(t-\tau(t)))]\,dt+g(t,x(t-\delta(t)))\,dW(t),\quad & t\geq 0,\\ x_0(\cdot)=\xi\in D^b_{{\mathcal F}_0}([m(0),0],H)\end{cases} \]where \(t-\tau (t),t-\delta(t)\to\infty\) with delays \(\tau (t),\delta(t)\to\infty\) as \(t\to \infty\). Our method for investigating the stability of solutions is based on the fixed point theorem. Cited in 25 Documents MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 93E03 Stochastic systems in control theory (general) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:mild solution; stochastic partial differential equation with delays; asymptotic stability PDFBibTeX XMLCite \textit{J. Luo}, J. Comput. Appl. Math. 222, No. 2, 364--371 (2008; Zbl 1151.60336) Full Text: DOI References: [1] J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint; J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint [2] Burton, T. A., Stability by Fixed Point Theory for Functional Differential Equations (2006), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 1090.45002 [3] Caraballo, T., Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 27-47 (1990) · Zbl 0723.60074 [4] Caraballo, T.; Liu, Kai, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stochastic Anal. Appl., 17, 743-763 (1999) · Zbl 0943.60050 [5] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge University Press · Zbl 0761.60052 [6] Govindan, T. E., Exponential stability in mean-square of parabolic quasilinear stochastic delay evolution equations, Stochastic Anal. Appl., 17, 443-461 (1999) · Zbl 0940.60076 [7] Haussmann, U. G., Asymptotic stability of the linear Itô equation in infinite dimensions, J. Math. Anal. Appl., 65, 219-235 (1978) · Zbl 0385.93051 [8] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl., 90, 12-44 (1982) · Zbl 0497.93055 [9] Liu, Kai, Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics, 63, 1-26 (1998) · Zbl 0947.93037 [10] Liu, Kai, (Stability of Infinite Dimensional Stochastic Differential Equations with Applications. Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Pitman Monographs Series in Pure and Applied Mathematics, vol. 135 (2005), Chapman & Hall/CRC) · Zbl 1085.60003 [11] Luo, Jiaowan, Fixed points and stability of neutral stochastic delay differential equations, J. Math. Anal. Appl. (2007) · Zbl 1160.60020 [12] Mao, Xuerong, Exponential stability for stochastic differential delay equations in Hilbert spaces, Quart. J. Math., 42, 77-85 (1991) · Zbl 0719.60062 [13] Taniguchi, T., Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics, 53, 41-52 (1995) · Zbl 0854.60051 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.