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 Mat. Sb., 1995, Volume 186, Number 1, Pages 29–46 (Mi msb2)

Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems

T. V. Girya, I. D. Chueshov

Kharkiv State University

Abstract: We prove the existence of inertial manifolds for a semilinear dynamical system perturbed by additive ‘white noise’. This manifold is generated by a certain predictable stationary vector process $\Phi_t(\omega)$. We study properties of this process as well as the properties of the induced finite-dimensional stochastic system on the manifold (inertial form). The results obtained allow us to prove for the original stochastic system a theorem on stabilization of stationary solutions to a unique invariant measure. This measure is uniquely defined by the probability distribution of the process $\Phi_t(\omega)$ and the form of the invariant measure corresponding to the inertial form.

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English version:
Sbornik: Mathematics, 1995, 186:1, 29–45

Bibliographic databases:

UDC: 517.919
MSC: 60G10, 34D45

Citation: T. V. Girya, I. D. Chueshov, “Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems”, Mat. Sb., 186:1 (1995), 29–46; Sb. Math., 186:1 (1995), 29–45

Citation in format AMSBIB
\Bibitem{GirChu95}
\by T.~V.~Girya, I.~D.~Chueshov
\paper Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems
\jour Mat. Sb.
\yr 1995
\vol 186
\issue 1
\pages 29--46
\mathnet{http://mi.mathnet.ru/msb2}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1641664}
\zmath{https://zbmath.org/?q=an:0851.60036}
\transl
\jour Sb. Math.
\yr 1995
\vol 186
\issue 1
\pages 29--45
\crossref{https://doi.org/10.1070/SM1995v186n01ABEH000002}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. D. Chueshov, “Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise”, J Dyn Diff Equat, 7:4 (1995), 549
2. Igor D. Chueshov, Pierre A. Vuillermot, “Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: itô's case”, Stochastic Analysis and Applications, 18:4 (2000), 581
3. S.I. Dudov, I.V. Zlatorunskaya, “Uniform estimate of a compact convex set by a ball in an arbitrary norm”, Sb. Math, 191:10 (2000), 1433
4. Jinqiao Duan, Kening Lu, Björn Schmalfuss, “Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations”, J Dyn Diff Equat, 16:4 (2004), 949
5. Chueshov, ID, “Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients”, Advanced Nonlinear Studies, 5:4 (2005), 461
6. Kening Lu, Björn Schmalfuß, “Invariant manifolds for stochastic wave equations”, Journal of Differential Equations, 236:2 (2007), 460
7. Björn Schmalfuss, Klaus R. Schneider, “Invariant Manifolds for Random Dynamical Systems with Slow and Fast Variables”, J Dyn Diff Equat, 20:1 (2008), 133
8. María J. Garrido-Atienza, Kening Lu, Björn Schmalfuß, “Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion”, Journal of Differential Equations, 248:7 (2010), 1637
9. Igor Chueshov, Björn Schmalfuß, “Master-slave synchronization and invariant manifolds for coupled stochastic systems”, J. Math. Phys, 51:10 (2010), 102702
10. Armen Shirikyan, Sergey Zelik, “Exponential attractors for random dynamical systems and applications”, Stoch PDE: Anal Comp, 2013
11. Wang W., Roberts A.J., “Macroscopic Reduction for Stochastic Reaction-Diffusion Equations”, IMA J. Appl. Math., 78:6 (2013), 1237–1264
12. Li J., Lu K., Bates P., “Normally Hyperbolic Invariant Manifolds for Random Dynamical Systems: Part I - Persistence”, Trans. Am. Math. Soc., 365:11 (2013), 5933–5966
13. Davit Martirosyan, “Exponential mixing for the white-forced damped nonlinear wave equation”, EECT, 3:4 (2014), 645
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