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doi: 10.3934/era.2021014

## Averaging principle on infinite intervals for stochastic ordinary differential equations

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 State University of Moldova, Faculty of Mathematics and Informatics, Department of Mathematics, A. Mateevich Street 60, MD–2009 Chişinǎu, Moldova

* Corresponding author: Zhenxin Liu

Dedicated to the memory of Professor Russell A. Johnson

Received  August 2020 Revised  January 2021 Published  February 2021

Fund Project: This work is partially supported by NSFC Grants 11522104, 11871132, 11925102, and Xinghai Jieqing and DUT19TD14 funds from Dalian University of Technology

In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.

Citation: David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, doi: 10.3934/era.2021014
##### References:
 [1] N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainskoǐ SSR, 1945. (in Russian)  Google Scholar [2] N. N. Bogolyubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Translated from the second revised Russian edition. International Monographs on Advanced Mathematics and Physics Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961.  Google Scholar [3] P. Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields, 83 (1989), 509-545.  doi: 10.1007/BF01845701.  Google Scholar [4] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.  Google Scholar [5] S. Cerrai and A. Lunardi, Averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations: the almost periodic case, SIAM J. Math. Anal., 49 (2017), 2843-2884.  doi: 10.1137/16M1063307.  Google Scholar [6] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009.  Google Scholar [7] D. N. Cheban, Global Attractors of Nonautonomous Dynamical and Control Systems, 2$^{nd}$ edition, Interdisciplinary Mathematical Sciences, vol.18, River Edge, NJ: World Scientific, 2015. doi: 10.1142/9297.  Google Scholar [8] D. Cheban and J. Duan, Recurrent motions and global attractors of non-autonomous Lorenz systems, Dyn. Syst., 19 (2004), 41–59. doi: 10.1080/14689360310001624132.  Google Scholar [9] D. Cheban and Z. Liu, Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations, J. Differential Equations, 269 (2020), 3652-3685.  doi: 10.1016/j.jde.2020.03.014.  Google Scholar [10] Ju. L. Dalec'kiǐ and M. G. Kreǐn, Stability of Solutions of Differential Equations in Banach Space, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 43. American Mathematical Society, Providence, R.I., 1974.  Google Scholar [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar [12] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights. Elsevier, Amsterdam, 2014. doi: 10.1016/B978-0-12-800882-9.00001-9.  Google Scholar [13] R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755347.  Google Scholar [14] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Translated from the 1979 Russian original by Joseph Szücs. 3$^rd$ edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.  Google Scholar [15] R. Z. Has'minskiǐ, On the principle of averaging the itô's stochastic differential equations, Kybernetika (Prague), 4 (1968), 260–279. (in Russian)  Google Scholar [16] M. Kamenskii, O. Mellah and P. Raynaud de Fitte, Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, J. Math. Anal. Appl., 427 (2015), 336-364.  doi: 10.1016/j.jmaa.2015.02.036.  Google Scholar [17] M. A. Krasnosel'skiǐ, V. Sh Burd and Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations, Nauka, Moscow, 1970 (in Russian). [English translation: Nonlinear Almost Periodic Oscillations. A Halsted Press Book. New York etc.: John Wiley & Sons; Jerusalem- London: Israel Program for Scientific Translations. IX, 326 p., 1973]  Google Scholar [18] N. Krylov and N. Bogolyubov, Introduction to Non-Linear Mechanics, Annals of Mathematics Studies, no. 11. Princeton University Press, Princeton, N. J., 1943.   Google Scholar [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow State University Press, Moscow, 1978.   Google Scholar [20] X. Liu and Z. Liu, Poisson stable solutions for stochastic differential equations with Lévy noise, Acta Math. Sin. (Engl. Ser.), In Press. (also available at arXiv: 2002.00395) Google Scholar [21] Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 226 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011.  Google Scholar [22] A. J. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, Phys. A, 387 (2008), 12-38.  doi: 10.1016/j.physa.2007.08.023.  Google Scholar [23] B. A. Ščerbakov, A certain class of Poisson stable solutions of differential equations, Differencial'nye Uravnenija, 4 (1968), 238–243. (in Russian)  Google Scholar [24] B. A. Ščerbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differencial'nye Uravnenija, 11 (1975), 1246–1255. (in Russian) [English translation: Differential Equations 11 (1975), 937–943].  Google Scholar [25] G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, 1971.  Google Scholar [26] B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Stiinca, Kishinev, 1972.  Google Scholar [27] B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Știinţa, Chişinǎu, 1985. (in Russian)  Google Scholar [28] K. S. Sibirsky, Introduction to Topological Dynamics, Kishinev, RIA AN MSSR, 1970. (in Russian). [English translation: Introduction to Topological Dynamics. Noordhoff, Leyden, 1975]  Google Scholar [29] A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translated from the Russian by H. H. McFaden. Translations of Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. doi: 10.1090/mmono/078.  Google Scholar [30] A. Yu. Veretennikov, On large deviations in the averaging principle for SDEs with a "full dependence'', Ann. Probab., 27 (1999), 284-296.  doi: 10.1214/aop/1022677263.  Google Scholar [31] I. Vrkoč, Weak averaging of stochastic evolution equations, Math. Bohem., 120 (1995), 91-111.  doi: 10.21136/MB.1995.125891.  Google Scholar [32] W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.  Google Scholar

show all references

##### References:
 [1] N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainskoǐ SSR, 1945. (in Russian)  Google Scholar [2] N. N. Bogolyubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Translated from the second revised Russian edition. International Monographs on Advanced Mathematics and Physics Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961.  Google Scholar [3] P. Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields, 83 (1989), 509-545.  doi: 10.1007/BF01845701.  Google Scholar [4] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.  Google Scholar [5] S. Cerrai and A. Lunardi, Averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations: the almost periodic case, SIAM J. Math. Anal., 49 (2017), 2843-2884.  doi: 10.1137/16M1063307.  Google Scholar [6] D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009.  Google Scholar [7] D. N. Cheban, Global Attractors of Nonautonomous Dynamical and Control Systems, 2$^{nd}$ edition, Interdisciplinary Mathematical Sciences, vol.18, River Edge, NJ: World Scientific, 2015. doi: 10.1142/9297.  Google Scholar [8] D. Cheban and J. Duan, Recurrent motions and global attractors of non-autonomous Lorenz systems, Dyn. Syst., 19 (2004), 41–59. doi: 10.1080/14689360310001624132.  Google Scholar [9] D. Cheban and Z. Liu, Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations, J. Differential Equations, 269 (2020), 3652-3685.  doi: 10.1016/j.jde.2020.03.014.  Google Scholar [10] Ju. L. Dalec'kiǐ and M. G. Kreǐn, Stability of Solutions of Differential Equations in Banach Space, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 43. American Mathematical Society, Providence, R.I., 1974.  Google Scholar [11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar [12] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights. Elsevier, Amsterdam, 2014. doi: 10.1016/B978-0-12-800882-9.00001-9.  Google Scholar [13] R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755347.  Google Scholar [14] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Translated from the 1979 Russian original by Joseph Szücs. 3$^rd$ edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.  Google Scholar [15] R. Z. Has'minskiǐ, On the principle of averaging the itô's stochastic differential equations, Kybernetika (Prague), 4 (1968), 260–279. (in Russian)  Google Scholar [16] M. Kamenskii, O. Mellah and P. Raynaud de Fitte, Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, J. Math. Anal. Appl., 427 (2015), 336-364.  doi: 10.1016/j.jmaa.2015.02.036.  Google Scholar [17] M. A. Krasnosel'skiǐ, V. Sh Burd and Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations, Nauka, Moscow, 1970 (in Russian). [English translation: Nonlinear Almost Periodic Oscillations. A Halsted Press Book. New York etc.: John Wiley & Sons; Jerusalem- London: Israel Program for Scientific Translations. IX, 326 p., 1973]  Google Scholar [18] N. Krylov and N. Bogolyubov, Introduction to Non-Linear Mechanics, Annals of Mathematics Studies, no. 11. Princeton University Press, Princeton, N. J., 1943.   Google Scholar [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow State University Press, Moscow, 1978.   Google Scholar [20] X. Liu and Z. Liu, Poisson stable solutions for stochastic differential equations with Lévy noise, Acta Math. Sin. (Engl. Ser.), In Press. (also available at arXiv: 2002.00395) Google Scholar [21] Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 226 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011.  Google Scholar [22] A. J. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, Phys. A, 387 (2008), 12-38.  doi: 10.1016/j.physa.2007.08.023.  Google Scholar [23] B. A. Ščerbakov, A certain class of Poisson stable solutions of differential equations, Differencial'nye Uravnenija, 4 (1968), 238–243. (in Russian)  Google Scholar [24] B. A. Ščerbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differencial'nye Uravnenija, 11 (1975), 1246–1255. (in Russian) [English translation: Differential Equations 11 (1975), 937–943].  Google Scholar [25] G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, 1971.  Google Scholar [26] B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Stiinca, Kishinev, 1972.  Google Scholar [27] B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Știinţa, Chişinǎu, 1985. (in Russian)  Google Scholar [28] K. S. Sibirsky, Introduction to Topological Dynamics, Kishinev, RIA AN MSSR, 1970. (in Russian). [English translation: Introduction to Topological Dynamics. Noordhoff, Leyden, 1975]  Google Scholar [29] A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translated from the Russian by H. H. McFaden. Translations of Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. doi: 10.1090/mmono/078.  Google Scholar [30] A. Yu. Veretennikov, On large deviations in the averaging principle for SDEs with a "full dependence'', Ann. Probab., 27 (1999), 284-296.  doi: 10.1214/aop/1022677263.  Google Scholar [31] I. Vrkoč, Weak averaging of stochastic evolution equations, Math. Bohem., 120 (1995), 91-111.  doi: 10.21136/MB.1995.125891.  Google Scholar [32] W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.  Google Scholar
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