Average and deviation for slow-fast stochastic partial differential equations

Details Average and deviation for slow-fast stochastic partial differential equations Average and deviation for slow-fast stochastic partial differential equations

2009-04-09

arxiv.org/abs/0904.1462v1

Mathematics

Analysis of PDEs

22 pages, 5 figures

Scientific paper

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 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 $\mathcal{O}(\e)$ instead of $\mathcal{O}(\sqrt{\e})$ attained in previous averaging.

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