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

Part of: Miscellaneous topics - Partial differential equations Stochastic analysis

Published online by Cambridge University Press:  28 September 2018

MANUEL V. GNANN ,
CHRISTIAN KUEHN Open the ORCID record for CHRISTIAN KUEHN [Opens in a new window]  and
ANNE PEIN
MANUEL V. GNANN
Affiliation:
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching Munich, Germany email: gnann@ma.tum.de, ckuehn@ma.tum.de, pein@ma.tum.de
CHRISTIAN KUEHN
Affiliation:
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching Munich, Germany email: gnann@ma.tum.de, ckuehn@ma.tum.de, pein@ma.tum.de
ANNE PEIN
Affiliation:
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching Munich, Germany email: gnann@ma.tum.de, ckuehn@ma.tum.de, pein@ma.tum.de
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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.

Keywords

Fast–slow systemstochastic partial differential equationsample pathcovariance neighbourhood

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 1004 - 1024
Copyright
© Cambridge University Press 2018 

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Footnotes

†

MVG received funding from the Deutsche Forschungsgemeinschaft (project number 334362478). CK and AP were supported by a Lichtenberg-Professorship awarded to CK. CK also acknowledges partial support via the DFG-DACH grant “Analysis of PDE with Cross-Diffusion and Stochastic Terms”. The authors would like to thank Dirk Blömker, Nils Berglund and Alexandra Neamtu for interesting discussions regarding fast–slow SPDEs and the reviewers for their helpful comments.

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