Stability of stochastic partial differential equations with infinite delays. (English) Zbl 1151.60336

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.


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)
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