Towards sample path estimates for fast–slow stochastic partial differential equations

Journal article (2019)

Authors

M.V. Gnann Technische Universität München
Christian Kuehn Technische Universität München
Anne Pein Technische Universität München
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Stochastic partial differential equation Fast-slow system Sample path Covariance neighborhood
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Published Date

2019-10

Language

English

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Abstract

Estimates for sample paths of fast–slow stochastic ordinary differential equations have become a key mathematical tool relevant for theory and applications. In particular, there have been breakthroughs by Berglund and Gentz to prove sharp exponential error estimates. In this paper, we take the first steps in order to generalise this theory to fast–slow stochastic partial differential equations. In a simplified setting with a natural decomposition into low- and high-frequency modes, we demonstrate that for a short-time period the probability for the corresponding sample path to leave a neighbourhood around the stable slow manifold of the system is exponentially small as well.