Skip to main content
×
×
Home

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

  • MANUEL V. GNANN (a1), CHRISTIAN KUEHN (a1) and ANNE PEIN (a1)
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.

Copyright
References
Hide All
[1]Benzi, R., Parisi, G., Sutera, A. & Vulpiani, A. (1982) Stochastic resonance in climatic change. Tellus 34(11), 1016.
[2]Berglund, N. & Gentz, B. (2002) Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theory Relat. Fields 3, 341388.
[3]Berglund, N. & Gentz, B. (2003) Geometric singular perturbation theory for stochastic differential equations. J. Differ. Equ. 191, 154.
[4]Berglund, N. & Gentz, B. (2006) Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer.
[5]Berglund, N. & Gentz, B. (2013) Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond. Electron. J. Probab. 18(24), 158.
[6]Berglund, N., Gentz, B. & Kuehn, C. (2015) From random Poincaré maps to stochastic mixed-mode-oscillation patterns. J. Dyn. Differ. Equ. 27(1), 83136.
[7]Berglund, N. & Kuehn, C. (2016) Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions. Electron. J. Probab. 21(18), 148.
[8]Blömker, D. (2007) Amplitude Equations for Stochastic Partial Differential Equations, World Scientific.
[9]Blömker, D. & Jentzen, A. (2013) Galerkin approximations for the stochastic Burgers equation. SIAM J. Numer. Anal. 51(1), 694715.
[10]Da Prato, G. & Zabczyk, J. (2014) Stochastic Equations in Infinite Dimensions, Cambridge University Press.
[11]Fenichel, N. (1979) Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 5398.
[12]Freidlin, M. I. & Wentzell, A. D. (1998) Random Perturbations of Dynamical Systems, Springer.
[13]Garcia-Ojalvo, J. & Sancho, J. (1999) Noise in Spatially Extended Systems, Springer.
[14]Gowda, K. & Kuehn, C. (2015) Warning signs for pattern-formation in SPDEs. Commun. Nonlinear Sci. Numer. Simul. 22(1), 5569.
[15]Hairer, M. (2014) A theory of regularity structures. Invent. Math. 198(2), 269504.
[16]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.
[17]Kuehn, C. (2015) Multiple Time Scale Dynamics, Springer.
[18]Kuehn, C. (2015) Numerical continuation and SPDE stability for the 2D cubic-quintic Allen-Cahn equation. SIAM/ASA J. Uncertainty Quantif. 3(1), 762789.
[19]Liaskos, K. B., Pantelous, A. A. & Stratis, I. G. (2015) Linear stochastic degenerate Sobolev equations and applications. Int. J. Control 88(12), 25382553.
[20]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.
[21]Lindner, B. & Schimansky-Geierc, L. (1999) Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance. Phys. Rev. E 60(6), 72707276.
[22]Majda, A. J., Timofeyev, I. & Vanden-Eijnden, E. (2001) A mathematical framework for stochastic climate models. Commun. Pure Appl. Math. 54, 891974.
[23]Van Neerven, J., Veraar, M. C. & Weis, L. (2008) Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255(4), 940993.
[24]Sadhu, S. & Kuehn, C. (2018) Stochastic mixed-mode oscillations in a three-species predator-prey model. Chaos 28(3), 033606.
[25]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.
[26]Su, J., Rubin, J. & Terman, D. (2004) Effects of noise on elliptic bursters. Nonlinearity 17, 133157.
[27]Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer.
[28]Veraar, M. C. (2010) Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations. J. Evol. Equ. 10(1), 85127.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed