Large Deviations and Averaging for Systems of Slow-Fast Stochastic Reaction-Diffusion Equations
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.
W. Hu et al., "Large Deviations and Averaging for Systems of Slow-Fast Stochastic Reaction-Diffusion Equations," Stochastics and Partial Differential Equations: Analysis and Computations, vol. 7, pp. 808-874, Springer, Dec 2019.
The definitive version is available at https://doi.org/10.1007/s40072-019-00140-y
Mathematics and Statistics
Keywords and Phrases
Averaging principle; Large deviations; Optimal control; Stochastic reaction-diffusion equations; Weak convergence method
International Standard Serial Number (ISSN)
Article - Journal
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01 Dec 2019