The convergence rate of approximate center manifolds for stochastic evolution equations via a Wong-Zakai type approximation (Q6587532)

The author considers the center manifolds of the following stochastic differential equation in a separable Hilbert space \(H\): \[ du =(Au + F(u))dt + u\circ dW, \quad u(0) =u_0, \] along with the approximated system: \[ {\frac {du_\delta}{dt}} =Au_\delta + F(u_\delta ) + u_\delta \mathcal{G}_\delta (\theta_t \omega), \quad u(0) =u_0, \] where \(A\) is a generator of a \(C_0\)-semigroup on \(H\), \(F: H \to H\) is a Lipschitz operator, \(W\) is a two-sided real-valued Wiener process on a probability space \((\Omega, \mathcal{F}, P)\), \(\delta>0\) is a parameter. Given \(t\in \mathbb{R}\), \(\theta_t: \Omega \to \Omega\) is given by \[ \theta_t \omega (\cdot) = \omega (t +\cdot) -\omega (t), \quad \forall \omega \in \Omega, \] and \[\mathcal{G}_\delta (\omega) = {\frac {\omega (\delta)}{\delta}}, \quad \forall \omega \in \Omega. \]\N\NUnder a gap condition, the author first establishes the existence of random center manifolds for the above systems, and then prove as \(\delta \to 0\), the center manifolds of the approximate system converge to that of the original stochastic system with convergence rate \(\delta^\gamma\) for \(\gamma \in (0, {\frac 12})\).