We study the asymptotic properties of the trajectories of a discrete-time random dynamical system in an infinite-dimensional Hilbert space. Under some natural assumptions on the model, we establish a multiplica-tive ergodic theorem with an exponential rate of convergence. The assumptions are satisfied for a large class of parabolic PDEs, including the 2D Navier--Stokes and complex Ginzburg--Landau equations perturbed by a non-degenerate bounded random kick force. As a consequence of this er-godic theorem, we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system. In particular , we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible. The proof relies on a refined version of the uniform Feller property combined with some contraction and bootstrap arguments.

中文翻译：

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不可约随机动力系统的乘法遍历定理

我们研究了无限维希尔伯特空间中离散时间随机动力系统的轨迹的渐近特性。在模型的一些自然假设下，我们建立了一个具有指数收敛速度的乘法遍历定理。对于一大类抛物线偏微分方程，包括 2D Navier--Stokes 和复杂的 Ginzburg--Landau 方程，这些假设都得到满足，这些方程受到非退化有界随机踢力的扰动。由于这个er-godic定理，我们得出了一些关于潜在随机动力系统轨迹统计特性的新结果。特别是，我们在系统不是不可约的情况下，获得了占领措施的大偏差原理和压力函数的分析性。