Hausdorff Dimension of Random Attractors for a Stochastic Delayed Parabolic Equation in Banach Spaces

The main purpose of this paper is to give an upper bound of Hausdorff dimension of random attractors for a stochastic delayed parabolic equation in Banach spaces. The estimation of dimensions of random attractors are obtained by combining the squeezing property and a covering lemma of finite subspace of Banach spaces, which generalizes the method established in Hilbert spaces. Due to the lack of smooth inner product geometry structure, we adopt the state decomposition of phase space based on the exponential dichotomy of the linear deterministic part of the studied equations instead of orthogonal projectors with finite ranks used for stochastic partial differential equations. The obtained dimension of the random attractors depends only on the inner characteristics of the studied equation, such as spectrum of the linear part and the random Lipschitz constant of the nonlinear term, while not relating to the compact embedding of the phase space to another Banach space as the existing works did.