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On nonuniform dichotomy for stochastic skew-evolution semiflows in Hilbert spaces. (English) Zbl 1274.37044
Let \(X\) be a Hilbert space, \(\varphi _{t\geq s\geq 0}\) a stochastic evolution semiflow on \(\Omega \) and \(\Phi _{t\geq s\geq 0}\) a stochastic evolution cocycle of linear bounded operators on \(X\) over the semiflow \(\varphi \). Then the couple \((\varphi (t,s,\omega ),\Phi (t,s,\omega )x)\) is called a stochastic skew-evolution semiflow on \(X\times \Omega \). It is assumed that \(X\) splits into \(X=X_1(\omega )\oplus X_2(\omega )\), where \(X_1(\omega )\) and \(X_2(\omega )\) are a stable and an unstable space, respectively, \(\Pi _1(\omega )\), \(\Pi _2(\omega )\) denote the respective projections, and it is assumed that \(\Phi [X_k]\subseteq X_k(\varphi )\) and \(\Pi _k(\varphi )\Phi =\Phi \Pi _k\) hold for \(k=1,2\). The operators \(\Phi _{\Pi _k}=\Phi \Pi _k\) are considered and the notions of strong measurability in mean square, exponential growth in mean square, exponential decay in mean square, integrally stable in mean square and integrally unstable in mean square for \(\Phi \) are introduced. By the main result, if \(\Phi \) is strongly measurable in mean square, \(\Phi _{\Pi _1}\) has exponential growth in mean square, \(\Phi _{\Pi _2}\) has exponential decay in mean square and a certain integral condition is satisfied then the stochastic skew-evolution semiflow \((\varphi ,\Phi )\) has exponential dichotomy in mean square, i.e. the stochastic evolution cocycle \(\Phi _{\Pi _1}\) is exponentially stable in mean square and \(\Phi _{\Pi _2}\) is exponentially unstable in mean square, which in other words means that there exist \(\varphi \)-invariant random variables \(\alpha \geq 0\), \(\nu >\alpha \) and a tempered random variable \(N\geq 1\) such that \[ \mathbb {E}\,\| \Phi _{\Pi _1}(t,t_0,\omega )x\| ^2\leq N(\omega )e^{\alpha (\omega )t}e^{-\nu (\omega )(t-s)}\mathbb {E}\,\| \Phi _{\Pi _1}(s,t_0,\omega )x\| ^2 \] and \[ \mathbb {E}\,\| \Phi _{\Pi _2}(s,t_0,\omega )x\| ^2\leq N(\omega )e^{\alpha (\omega )t}e^{-\nu (\omega )(t-s)}\mathbb E\,\| \Phi _{\Pi _1}(t,t_0,\omega )x\| ^2 \] hold for every \(0\leq t_0\leq s\leq t\), \(\omega \in \Omega \) and \(x\in X\).
37L55 Infinite-dimensional random dynamical systems; stochastic equations
60H25 Random operators and equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
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