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

Published online by Cambridge University Press:  28 September 2018

Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching Munich, Germany email:,,
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching Munich, Germany email:,,
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching Munich, Germany email:,,


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.

© Cambridge University Press 2018 

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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.


Benzi, R., Parisi, G., Sutera, A. & Vulpiani, A. (1982) Stochastic resonance in climatic change. Tellus 34(11), 1016.CrossRefGoogle Scholar
Berglund, N. & Gentz, B. (2002) Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theory Relat. Fields 3, 341388.CrossRefGoogle Scholar
Berglund, N. & Gentz, B. (2003) Geometric singular perturbation theory for stochastic differential equations. J. Differ. Equ. 191, 154.CrossRefGoogle Scholar
Berglund, N. & Gentz, B. (2006) Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer.Google Scholar
Berglund, N. & Gentz, B. (2013) Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond. Electron. J. Probab. 18(24), 158.CrossRefGoogle Scholar
Berglund, N., Gentz, B. & Kuehn, C. (2015) From random Poincaré maps to stochastic mixed-mode-oscillation patterns. J. Dyn. Differ. Equ. 27(1), 83136.CrossRefGoogle Scholar
Berglund, N. & Kuehn, C. (2016) Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions. Electron. J. Probab. 21(18), 148.CrossRefGoogle Scholar
Blömker, D. (2007) Amplitude Equations for Stochastic Partial Differential Equations, World Scientific.CrossRefGoogle Scholar
Blömker, D. & Jentzen, A. (2013) Galerkin approximations for the stochastic Burgers equation. SIAM J. Numer. Anal. 51(1), 694715.CrossRefGoogle Scholar
Da Prato, G. & Zabczyk, J. (2014) Stochastic Equations in Infinite Dimensions, Cambridge University Press.CrossRefGoogle Scholar
Fenichel, N. (1979) Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 5398.CrossRefGoogle Scholar
Freidlin, M. I. & Wentzell, A. D. (1998) Random Perturbations of Dynamical Systems, Springer.CrossRefGoogle Scholar
Garcia-Ojalvo, J. & Sancho, J. (1999) Noise in Spatially Extended Systems, Springer.CrossRefGoogle Scholar
Gowda, K. & Kuehn, C. (2015) Warning signs for pattern-formation in SPDEs. Commun. Nonlinear Sci. Numer. Simul. 22(1), 5569.CrossRefGoogle Scholar
Hairer, M. (2014) A theory of regularity structures. Invent. Math. 198(2), 269504.CrossRefGoogle Scholar
Jones, C. K. R. T. (1995) Geometric singular perturbation theory. In: Dynamical Systems (Montecatini Terme, 1994), volume 1609 of Lecture Notes in Mathematics, Springer, pp. 44118.CrossRefGoogle Scholar
Kuehn, C. (2015) Multiple Time Scale Dynamics, Springer.CrossRefGoogle Scholar
Kuehn, C. (2015) Numerical continuation and SPDE stability for the 2D cubic-quintic Allen-Cahn equation. SIAM/ASA J. Uncertainty Quantif. 3(1), 762789.CrossRefGoogle Scholar
Liaskos, K. B., Pantelous, A. A. & Stratis, I. G. (2015) Linear stochastic degenerate Sobolev equations and applications. Int. J. Control 88(12), 25382553.CrossRefGoogle Scholar
Liaskos, K. B., Stratis, I. G. & Pantelous, A. A. (2018) Stochastic degenerate Sobolev equations: well posedness and exact controllability. Math. Methods Appl. Sci. 41(3), 10251032.CrossRefGoogle Scholar
Lindner, B. & Schimansky-Geierc, L. (1999) Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance. Phys. Rev. E 60(6), 72707276.CrossRefGoogle ScholarPubMed
Majda, A. J., Timofeyev, I. & Vanden-Eijnden, E. (2001) A mathematical framework for stochastic climate models. Commun. Pure Appl. Math. 54, 891974.CrossRefGoogle Scholar
Van Neerven, J., Veraar, M. C. & Weis, L. (2008) Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255(4), 940993.CrossRefGoogle Scholar
Sadhu, S. & Kuehn, C. (2018) Stochastic mixed-mode oscillations in a three-species predator-prey model. Chaos 28(3), 033606.CrossRefGoogle Scholar
Sieber, M., Malchow, H. & Schimansky-Geier, L. (2007) Constructive effects of environmental noise in an excitable prey-predator plankton system with infected prey. Ecol. Complex. 4(4), 223233.CrossRefGoogle Scholar
Su, J., Rubin, J. & Terman, D. (2004) Effects of noise on elliptic bursters. Nonlinearity 17, 133157.CrossRefGoogle Scholar
Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer.CrossRefGoogle Scholar
Veraar, M. C. (2010) Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations. J. Evol. Equ. 10(1), 85127.CrossRefGoogle Scholar