AI 中文总结
研究可分希尔伯特空间中一类耗散随机演化方程长时间小噪声行为，建立占据测度大偏差原理，涵盖多种无穷维方程。通过解析半群等技术，结合弱收敛方法，给出速率函数显式公式并构建近最优控制。
AI 中文摘要
我们研究了一类在可分希尔伯特空间中的耗散随机演化方程的长时间小噪声行为，该方程由圆柱维纳过程驱动，其协方差在强算子拓扑中退化为极限算子。一个典型例子是有界域上的随机反应扩散方程，其噪声在空间上均匀但频谱正则化，在极限情况下变得空间粗糙。我们建立了随着时间范围变大、噪声强度趋于零且噪声变得越来越粗糙时占据测度的大偏差原理。结果涵盖了广泛的无穷维耗散方程，包括由渐近粗糙圆柱噪声驱动的方程，其协方差不必是迹类。扩展了Budhiraja和Zoubouloglou的有限维工作，无穷维设置引入了新的困难：有界论证需要仔细处理无界演化和逆协方差算子，噪声粗糙度增加必须通过一致估计与幅度消失相平衡。证明结合了解析半群技术、分数域空间估计以及加权空间中随机卷积的仔细处理。速率函数由一个简单显式公式给出，即每个点消除确定性漂移的卡梅伦 - 马丁成本平方的测度平均值。证明遵循基于Bouè - Dupuis变分公式的弱收敛方法，并通过交替旅行阶段构建近最优控制，这些阶段在规定目标状态之间引导过程，以及保持阶段，在塑造占据测度时将其稳定在目标附近。
英文摘要
We study the long-time, small-noise behavior of a class of dissipative stochastic evolution equations in a separable Hilbert space, driven by a cylindrical Wiener process whose covariance degenerates to a limiting operator in the strong operator topology. A prototypical example is a stochastic reaction-diffusion equation on a bounded domain with spatially homogeneous but spectrally regularized noise that becomes spatially rough in the limit. We establish a large deviation principle for occupation measures as the time horizon becomes large, the noise intensity tends to zero, and the noise becomes increasingly rough. The result covers a broad class of dissipative equations in infinite dimensions, including those driven by asymptotically rough cylindrical noise whose covariance need not be trace class. Extending the finite-dimensional work of Budhiraja and Zoubouloglou, the infinite-dimensional setting introduces substantial new difficulties: the bound arguments require careful handling of the unbounded evolution and inverse covariance operators, and the increasing roughness of the noise must be balanced against its vanishing amplitude through uniform estimates. Proofs combine analytic semigroup techniques, fractional domain space estimates and a careful treatment of stochastic convolutions in weighted spaces. The rate function is given by a simple explicit formula, the average over the measure of the squared Cameron-Martin cost of canceling the deterministic drift at each point. The proof follows the weak convergence approach based on the Bouè-Dupuis variational formula and constructs near-optimal controls by alternating travel phases that steer the process between prescribed target states and hold phases that stabilize it near a target while shaping the occupation measure.
