We study a large deviation principle for a system of stochastic reaction–diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the fast process and the associated control that arises from the weak convergence method decouple from each other. We show that in this decoupling case one can use the weak convergence method to characterize the limiting process via a “viable pair” that captures the limiting controlled dynamics and the effective invariant measure simultaneously. The characterization of the limit of the controlled slow-fast processes in terms of viable pair enables us to obtain a variational representation of the large deviation action functional. Due to the infinite-dimensional nature of our set-up, the proof of tightness as well as the analysis of the limit process and in particular the proof of the large deviations lower bound is considerably more delicate here than in the finite-dimensional situation. Smoothness properties of optimal controls in infinite dimensions (a necessary step for the large deviations lower bound) need to be established. We emphasize that many issues that are present in the infinite dimensional case, are completely absent in finite dimensions.

中文翻译：

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慢-快随机反应-扩散方程组的大偏差和平均

我们研究了随机反应-扩散方程（SRDE）系统的大偏差原理，该系统将快慢分量分开，慢速分量中的噪声很小。大偏差原理的推导基于无限维的弱收敛方法，这导致对受控SRDE的平均值进行研究。通过适当地选择参数，由弱收敛方法引起的快速过程和相关控制彼此分离。我们表明，在这种解耦情况下，人们可以使用弱收敛方法通过“可行对”来表征极限过程，该“可行对”同时捕获极限受控动力学和有效不变量度。根据可行对对受控慢速-快速过程的限制进行表征，使我们能够获得大偏差动作功能的变化表示。由于我们的装置具有无限尺寸的性质，因此与有限尺寸的情况相比，密封性的证明以及对极限过程的分析，尤其是大偏差下限的证明在这里要微妙得多。需要建立无限尺寸的最佳控制的平滑特性（对于大偏差下限而言是必要的步骤）。我们强调，在无限维情况下存在的许多问题在有限维中是完全不存在的。与有限尺寸的情况相比，密封性的证明以及对极限过程的分析，尤其是大偏差下限的证明，在这里要微妙得多。需要建立无限尺寸的最佳控制的平滑特性（对于大偏差下限而言是必要的步骤）。我们强调，在无限维情况下存在的许多问题在有限维中是完全不存在的。与有限尺寸的情况相比，密封性的证明以及对极限过程的分析，尤其是大偏差下限的证明，在这里要微妙得多。需要建立无限尺寸的最佳控制的平滑特性（对于大偏差下限而言是必要的步骤）。我们强调，在无限维情况下存在的许多问题在有限维中是完全不存在的。