Random attractors for stochastic evolution equations driven by fractional Brownian motion (Q2927853)

An evolution equation \(du=Au\,dt+G(u)\,d\omega\), \(u(0)=u_0\), is considered on a separable Hilbert space \(V\). Here, \(-A\) is a strictly positive, symmetric operator with a compact inverse such that \(A\) is an infinitesimal generator of an analytic and exponentially decreasing semigroup on \(V\), \(G\) is a \(C^2\)-smooth mapping from \(V\) to the space of Hilbert-Schmidt operators on \(V\) with \(DG\) and \(D^2G\) bounded and \(\omega\) is a \(V\)-valued Hölder continuous path with the Hölder exponent in \((1/2,1)\). Integrals with respect to \(d\omega\) are understood in the sense of Zähle's generalization of the Young integral. Let the set of locally \(\beta^{\prime\prime}\)-Hölder-continuous functions \(\omega:\mathbb R\to V\), \(\omega(0)=0\), for some \(\beta^{\prime\prime}\in(1/2,1)\) be denoted by \(\Omega\) and consider the flow \(\theta_t\omega=\omega(t+\cdot)-\omega(t)\). It is shown that the mild solutions \(\varphi(\cdot,\omega,u_0)\) of the above equation define a nonautonomous dynamical system on \(V\). Using suitable (non-Markovian) stopping times \(\{T_i(\omega)\}_{i\in\mathbb Z}\) that satisfy a cocycle property for \((\theta_t)\), the authors consider a discrete nonautonomous dynamical system \((\tilde{\theta},\Phi)\) on \(V\), derived from \((\theta_t,\varphi)\), and prove that, under certain conditions, it has a unique pullback attractor with respect to a system of backward \(\nu\)-exponentially growing sets in \(V\). In the next section, attracting sets for \(\varphi\) are studied in the discrete as well as in the time-continuous case. In the last section, the case when \(\omega\) is a fractional Brownian motion in \(V\) with the Hurst parameter \(H\in(1/2,1)\) and a spatial covariance \(Q\) is studied. It is proved that the system under consideration defines an ergodic, metric dynamical system and the (pathwise) mild solutions define a random dynamical system, for which the pullback attractor constructed in the previous section is also a unique random attractor attracting random tempered sets.