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Normally hyperbolic invariant manifolds for random dynamical systems. I: Persistence. (English) Zbl 1291.37093
The authors consider a $$C^r$$ flow $$\psi(t)$$ in $$\mathbb R^n$$ with a compact and connected $$C^r$$ normally hyperbolic invariant manifold $$\mathcal M$$, and a random dynamical system $$\phi(t,\omega)$$ over a probabilistic space $$(\Omega,\mathcal F, P)$$ to be its randomly perturbed counterpart. The main result states that if the random flow is close to the deterministic one, i.e., there exists $$\rho>0$$ such that if $$\|\phi(t,\omega)-\psi(t)\|_{C^1}< \rho$$ for all $$\omega\in\Omega$$ and $$t\in[0,1]$$, then it also has a $$C^1$$ normally hyperbolic “random invariant manifold” $$\mathcal{\tilde M} (\omega)$$, which, if the hyperbolicity of the deterministic flow is strong enough, is a $$C^r$$ manifold diffeomorphic to $$\mathcal M$$. Moreover, the existence of stable and unstable manifolds for $$\phi(\omega,t)$$ at $$\mathcal{\tilde M} (\omega)$$ is proved.
In addition, the authors formulate a similar result on the persistence for flows with normally hyperbolic overflowing or inflowing invariant manifolds and its random counterpart.
The authors announce the paper as the first step in a program to build geometric singular perturbation theory under stochastic perturbations, by providing the persistence theory for random normally hyperbolic invariant manifolds. This work was already followed by [“Invariant foliations for random dynamical systems”, Discrete Contin. Dyn. Syst., Ser. A, 34 (9), 3639–3666 (2014)], where the same authors prove the existence of invariant foliations of stable and unstable manifolds of a normally hyperbolic random invariant manifold.

##### MSC:
 37H10 Generation, random and stochastic difference and differential equations 34F05 Ordinary differential equations and systems with randomness 34C45 Invariant manifolds for ordinary differential equations 37D10 Invariant manifold theory for dynamical systems 37D05 Dynamical systems with hyperbolic orbits and sets
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##### References:
 [1] Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. · Zbl 0906.34001 [2] P. W. Bates, P. C. Fife, R. A. Gardner, and C. K. R. T. Jones, Phase field models for hypercooled solidification, Phys. D 104 (1997), no. 1, 1 – 31. · Zbl 0890.35161 · doi:10.1016/S0167-2789(96)00207-2 · doi.org [3] Peter W. Bates, Kening Lu, and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), no. 645, viii+129. · Zbl 1023.37013 · doi:10.1090/memo/0645 · doi.org [4] Peter W. Bates, Kening Lu, and Chongchun Zeng, Persistence of overflowing manifolds for semiflow, Comm. Pure Appl. Math. 52 (1999), no. 8, 983 – 1046. , https://doi.org/10.1002/(SICI)1097-0312(199908)52:83.3.CO;2-F · Zbl 0947.37009 [5] Peter W. Bates, Kening Lu, and Chongchun Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4641 – 4676. · Zbl 0964.37018 [6] Alain Bensoussan and Franco Flandoli, Stochastic inertial manifold, Stochastics Stochastics Rep. 53 (1995), no. 1-2, 13 – 39. · Zbl 0854.60059 [7] Petra Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields 83 (1989), no. 4, 509 – 545. · Zbl 0671.58045 · doi:10.1007/BF01845701 · doi.org [8] M. Brin and Yu. Kifer, Dynamics of Markov chains and stable manifolds for random diffeomorphisms, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 351 – 374. · Zbl 0656.58033 · doi:10.1017/S0143385700004107 · doi.org [9] P. Brunovský, Tracking invariant manifolds without differential forms, Acta Math. Univ. Comenian. (N.S.) 65 (1996), no. 1, 23 – 32. · Zbl 0879.34057 [10] Tomás Caraballo, Jinqiao Duan, Kening Lu, and Björn Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud. 10 (2010), no. 1, 23 – 52. · Zbl 1209.37094 · doi:10.1515/ans-2010-0102 · doi.org [11] Tomás Caraballo, José A. Langa, and James C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2013, 2041 – 2061. · Zbl 0996.60070 · doi:10.1098/rspa.2001.0819 · doi.org [12] Andrew Carverhill, A formula for the Lyapunov numbers of a stochastic flow. Application to a perturbation theorem, Stochastics 14 (1985), no. 3, 209 – 226. · Zbl 0557.60048 · doi:10.1080/17442508508833339 · doi.org [13] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. · Zbl 0346.46038 [14] Giuseppe Da Prato and Arnaud Debussche, Construction of stochastic inertial manifolds using backward integration, Stochastics Stochastics Rep. 59 (1996), no. 3-4, 305 – 324. · Zbl 0876.60040 [15] Stephan Dahlke, Invariant manifolds for products of random diffeomorphisms, J. Dynam. Differential Equations 9 (1997), no. 2, 157 – 210. · Zbl 0886.58058 · doi:10.1007/BF02219220 · doi.org [16] Pavel Drábek and Jaroslav Milota, Methods of nonlinear analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. Applications to differential equations. · Zbl 1176.35002 [17] Jinqiao Duan, Kening Lu, and Björn Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab. 31 (2003), no. 4, 2109 – 2135. · Zbl 1052.60048 · doi:10.1214/aop/1068646380 · doi.org [18] Jinqiao Duan, Kening Lu, and Björn Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations 16 (2004), no. 4, 949 – 972. · Zbl 1065.60077 · doi:10.1007/s10884-004-7830-z · doi.org [19] Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/1972), 193 – 226. · Zbl 0246.58015 · doi:10.1512/iumj.1971.21.21017 · doi.org [20] Neil Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1973/74), 1109 – 1137. · Zbl 0284.58008 · doi:10.1512/iumj.1974.23.23090 · doi.org [21] Neil Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J. 26 (1977), no. 1, 81 – 93. · Zbl 0365.58012 · doi:10.1512/iumj.1977.26.26006 · doi.org [22] María J. Garrido-Atienza, Kening Lu, and Björn Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations 248 (2010), no. 7, 1637 – 1667. · Zbl 1186.37094 · doi:10.1016/j.jde.2009.11.006 · doi.org [23] T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Mat. Sb. 186 (1995), no. 1, 29 – 46 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 1, 29 – 45. · Zbl 0851.60036 · doi:10.1070/SM1995v186n01ABEH000002 · doi.org [24] J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901), 224-228. · JFM 32.0314.01 [25] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. · Zbl 0355.58009 [26] C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), no. 1, 64 – 88. · Zbl 0796.34038 · doi:10.1006/jdeq.1994.1025 · doi.org [27] Christopher K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc. 286 (1984), no. 2, 431 – 469. · Zbl 0567.35044 [28] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0148.12601 [29] Martin Krupa, Björn Sandstede, and Peter Szmolyan, Fast and slow waves in the FitzHugh-Nagumo equation, J. Differential Equations 133 (1997), no. 1, 49 – 97. · Zbl 0898.34050 · doi:10.1006/jdeq.1996.3198 · doi.org [30] Weigu Li and Kening Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math. 58 (2005), no. 7, 941 – 988. · Zbl 1075.37015 · doi:10.1002/cpa.20083 · doi.org [31] Zeng Lian and Kening Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc. 206 (2010), no. 967, vi+106. · Zbl 1200.37047 · doi:10.1090/S0065-9266-10-00574-0 · doi.org [32] Pei-Dong Liu and Min Qian, Smooth ergodic theory of random dynamical systems, Lecture Notes in Mathematics, vol. 1606, Springer-Verlag, Berlin, 1995. · Zbl 0841.58041 [33] Kening Lu and Björn Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differential Equations 236 (2007), no. 2, 460 – 492. · Zbl 1113.37056 · doi:10.1016/j.jde.2006.09.024 · doi.org [34] Salah-Eldin A. Mohammed and Michael K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab. 27 (1999), no. 2, 615 – 652. · Zbl 0940.60084 · doi:10.1214/aop/1022677380 · doi.org [35] Salah-Eldin A. Mohammed, Tusheng Zhang, and Huaizhong Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc. 196 (2008), no. 917, vi+105. · Zbl 1169.60014 · doi:10.1090/memo/0917 · doi.org [36] David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243 – 290. · Zbl 0493.58015 · doi:10.2307/1971392 · doi.org [37] Stephen Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equations 205 (2004), no. 1, 185 – 210. · Zbl 1077.35095 · doi:10.1016/j.jde.2004.06.013 · doi.org [38] Stephen Schecter, Exchange lemmas. II. General exchange lemma, J. Differential Equations 245 (2008), no. 2, 411 – 441. · Zbl 1158.34038 · doi:10.1016/j.jde.2007.10.021 · doi.org [39] Björn Schmalfuss, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl. 225 (1998), no. 1, 91 – 113. · Zbl 0931.37019 · doi:10.1006/jmaa.1998.6008 · doi.org [40] Wei Wang and Jinqiao Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys. 48 (2007), no. 10, 102701, 14. · Zbl 1152.81629 · doi:10.1063/1.2800164 · doi.org [41] Thomas Wanner, Linearization of random dynamical systems, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 203 – 269. · Zbl 0824.34069 [42] Hassler Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645 – 680. · Zbl 0015.32001 · doi:10.2307/1968482 · doi.org
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