Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory

Authors:
Janusz Mierczyński and Wenxian Shen

Journal:
Trans. Amer. Math. Soc. **365** (2013), 5329-5365

MSC (2010):
Primary 37H15, 37L55, 37A30; Secondary 15B52, 34F05, 35R60

DOI:
https://doi.org/10.1090/S0002-9947-2013-05814-X

Published electronically:
March 26, 2013

MathSciNet review:
3074376

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Abstract: This series of papers is concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part of the series focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of principal Floquet subspaces, principal Lyapunov exponents, and exponential separations for strongly positive deterministic systems in strongly ordered Banach to general positive random dynamical systems in ordered Banach spaces. Under some quite general assumptions, it is then shown that a positive random dynamical system in an ordered Banach space admits a family of generalized principal Floquet subspaces, a generalized principal Lyapunov exponent, and a generalized exponential separation. We will consider in the forthcoming part(s) the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.

**1.**L. Arnold, ``Random Dynamical Systems'', Springer Monogr. Math., Springer, Berlin, 1998. MR**2000m:37087****2.**L. Arnold, V. M. Gundlach and L. Demetrius,*Evolutionary formalism for products of positive random matrices*, Ann. Appl. Probab.**4**(1994), no. 3, 859-901. MR**95h:28028****3.**I. Chueshov, ``Monotone Random Systems Theory and Applications'', Lecture Notes in Math.,**1779**, Springer, Berlin, 2002. MR**2003d:37072****4.**J. Duan, K. Lu and B. Schmalfuß,*Invariant manifolds for stochastic partial differential equations*, Ann. Probab.**31**(2003), no. 4, 2109-2135. MR**2004m:60136****5.**S. P. Eveson,*Hilbert's projective metric and the spectral properties of positive linear operators*, Proc. London Math. Soc. (3)**70**(1995), no. 2, 411-440. MR**96g:47030****6.**J. Húska,*Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains*, J. Differential Equations**226**(2006), no. 2, 541-557. MR**2007h:35144****7.**J. Húska and P. Poláčik,*The principal Floquet bundle and exponential separation for linear parabolic equations*, J. Dynam. Differential Equations**16**(2004), no. 2, 347-375. MR**2006e:35147****8.**J. Húska, P. Poláčik and M. V. Safonov,*Harnack inequality, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations*, Ann. Inst. H. Poincaré Anal. Non Linéaire**24**(2007), no. 5, 711-739. MR**2008k:35211****9.**V. Hutson, W. Shen and G. T. Vickers,*Estimates for the principal spectrum point for certain time-dependent parabolic operators*, Proc. Amer. Math. Soc.**129**(2000), no. 6, 1669-1679. MR**2001m:35243****10.**R. Johnson, K. Palmer and G. R. Sell,*Ergodic properties of linear dynamical systems*, SIAM J. Math. Anal.**18**(1987), no. 1, 1-33. MR**88a:58112****11.**M. S. Keener and C. C. Travis,*Positive cones and focal points for a class of th order differential equations*, Trans. Amer. Math. Soc.**237**(1978), 331-351. MR**80i:34050****12.**M. A. Krasnoselskiĭ, ``Positive Solutions of Operator Equations'', translated from the Russian by R. E. Flaherty; edited by L. F. Boron, P. Noordhoff Ltd., Groningen, 1964. MR**31:6107****13.**U. Krengel, ``Ergodic Theorems'', Walter de Gruyter, Berlin, 1985. MR**87i:28001****14.**Z. Lian and K. Lu, ``Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems on a Banach Space'', Mem. Amer. Math. Soc.**206**(2010), no. 967. MR**2011g:37145****15.**C. Liverani,*Decay of correlations*, Ann. of Math. (2)**142**(1995), no. 2, 239-301. MR**96e:58090****16.**M. A. Lyapunov,*The general problem of the stability of motion*, translated by A. T. Fuller from É. Davaux's French translation (1907) of the 1892 Russian original, with an editorial (historical introduction) by Fuller, a biography of Lyapunov by V. I. Smirnov, and the bibliography of Lyapunov's works collected by J. F. Barrett, in: Lyapunov centenary issue, Internat. J. Control**55**(1992), no. 3, 521-790. MR**93e:01035****17.**R. Mañé, ``Ergodic Theory and Differentiable Dynamics'', translated from the Portuguese by S. Levy, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1987. MR**88c:58040****18.**J. Mierczyński,*Globally positive solutions of linear parabolic PDEs of second order with Robin boundary conditions*, J. Math. Anal. Appl.**209**(1997), no. 1, 47-59. MR**98c:35071****19.**-,*Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions*, J. Math. Anal. Appl.**226**(1998), no. 2, 326-347. MR**99m:35096****20.**-,*The principal spectrum for linear nonautonomous parabolic PDEs of second order: Basic properties*, in: Special issue in celebration of Jack K. Hale's 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998), J. Differential Equations**168**(2000), no. 2, 453-476. MR**2001m:35147****21.**J. Mierczyński and W. Shen,*Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations*, J. Differential Equations**191**(2003), no. 1, 175-205. MR**2004h:35232****22.**-,*Time averaging for nonautonomous/random parabolic equations*, Discrete Contin. Dyn. Syst. Ser. B**9**(2008), no. 3/4, 661-699. MR**2009a:35108****23.**-, ``Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications'', Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math.,**139**, CRC Press, Boca Raton, FL, 2008. MR**2010g:35216****24.**R. D. Nussbaum, ``Hilbert's Projective Metric and Iterated Nonlinear Maps'', Mem. Amer. Math. Soc.**75**(1988), no. 391. MR**89m:47046****25.**V. M. Millionshchikov,*Metric theory of linear systems of differential equations*, Math. USSR-Sb.**6**(1968), 149-158. MR**38:383****26.**V. I. Oseledets,*A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems*, Trans. Moscow. Math. Soc.**19**(1968), 197-231. MR**39:1629****27.**P. Poláčik,*On uniqueness of positive entire solutions and other properties of linear parabolic equations*, Discrete Contin. Dyn. Syst.**12**(2005), no. 1, 13-26. MR**2005k:35170****28.**P. Poláčik and I. Tereščák,*Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems*, Arch. Rational Mech. Anal.**116**(1991), 339-360. MR**93b:58088****29.**-,*Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations*, J. Dynam. Differential Equations**5**(1993), no. 2, 279-303. MR**94d:47064****30.**M. S. Raghunathan,*A proof of Oseledec's multiplicative ergodic theorem*, Israel J. Math.**32**(1979), no. 4, 356-362. MR**81f:60016****31.**D. Ruelle,*Ergodic theory of differentiable dynamical systems*, Inst. Hautes Ètudes Sci. Publ. Math. No.**50**(1979), 27-58. MR**81f:58031****32.**-,*Analycity properties of the characteristic exponents of random matrix products*, Adv. in Math.**32**(1979), no. 1, 68-80. MR**80e:58035****33.**-,*Characteristic exponents and invariant manifolds in Hilbert space*, Ann. of Math. (2)**115**(1982), no. 2, 243-290. MR**83j:58097****34.**H. H. Schaefer, ``Topological Vector Spaces'', fourth printing corrected, Grad. Texts in Math., Vol.**3**, Springer, New York-Berlin, 1980. MR**49:7722****35.**K.-U. Schaumlöffel and F. Flandoli,*A multiplicative ergodic theorem with applications to a first order stochastic hyperbolic equation in a bounded domain*, Stochastics Stochastics Rep.**34**(1991), no. 3-4, 241-255. MR**92m:60050****36.**Š. Schwabik and G. Ye, ``Topics in Banach Space Integration'', Ser. Real Anal.,**10**, World Scientific, Hackensack, NJ, 2005. MR**2006g:28002****37.**W. Shen and G. T. Vickers,*Spectral theory for general nonautonomous/random dispersal evolution operators*, J. Differential Equations**235**(2007), no. 1, 262-297. MR**2008d:35091****38.**W. Shen and Y. Yi, ``Almost Automorphic and Almost Periodic Dynamics in Skew-product Semiflows, Part II. Skew-product Semiflows'', Mem. Amer. Math. Soc.**136**(1998), no. 647. MR**99d:34088**

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Additional Information

**Janusz Mierczyński**

Affiliation:
Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland

Email:
mierczyn@pwr.wroc.pl

**Wenxian Shen**

Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849

Email:
wenxish@auburn.edu

DOI:
https://doi.org/10.1090/S0002-9947-2013-05814-X

Keywords:
Random dynamical system,
ordered Banach space,
principal Lyapunov exponent,
principal Floquet subspaces,
exponential separation,
entire positive solution,
multiplicative ergodic theorem,
Hilbert projective metric,
skew-product semiflow

Received by editor(s):
February 16, 2011

Received by editor(s) in revised form:
February 11, 2012

Published electronically:
March 26, 2013

Additional Notes:
The first author was supported by Resources for Science in years 2009-2012 as research project (grant MENII N N201 394537, Poland)

The second author was partially supported by NSF grant DMS-0907752

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.