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Average and deviation for slow-fast stochastic partial differential equations. (English) Zbl 1251.35201
Summary: Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the stochastic deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of order $$\mathcal O(\epsilon )$$ instead of order $$\mathcal O(\sqrt \epsilon)$$ attained in previous averaging.

##### MSC:
 35R60 PDEs with randomness, stochastic partial differential equations 60G44 Martingales with continuous parameter 60B10 Convergence of probability measures
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