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Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations. (English) Zbl 1444.34070

This paper studies the semi-linear stochastic differential equation (SDE) \[ dx(t)=(Ax(t)+f(t,x(t)))dt +g(t,x(t))dW(t) \tag{1} \] where \(f,g\) are Poisson stable functions in \(t\), and \(A\) is an exponentially stable linear operator.
Conditions on \(f\) and \(g\) are specified that lead to a proof that SDE (1) has a unique bounded solution. It is established that if in distribution \(f\) and \(g\) are jointly stationary (respectively, tau-periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable) uniformly in \(t\) with respect to \(x\) on every bounded subset, then so is the solution of SDE (1). In addition, it is shown that if in distribution \(f\) and \(g\) are also pseudo-periodic (resp. pseudo-recurrent), then so is the solution of SDE (1). Further theorems are proved that give conditions that imply that the solution of SDE (1) is globally asymptotically stable. The paper is concluded by presenting two examples giving applications.

MSC:

34F05 Ordinary differential equations and systems with randomness
34C25 Periodic solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34D20 Stability of solutions to ordinary differential equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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[1] Arnold, L.; Tudor, C., Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stoch. Stoch. Rep., 64, 177-193 (1998) · Zbl 1043.60513
[2] Basit, B.; Gnzler, H., Spectral criteria for solutions of evolution equations and comments on reduced spectra (2010), preprint
[3] Bebutov, V. M., On the shift dynamical systems on the space of continuous functions, Bull. of Inst. of Math. of Moscow University, 2, 5, 1-52 (1941), (in Russian)
[4] Bezandry, P. H.; Diagana, T., Existence of almost periodic solutions to some stochastic differential equations, Appl. Anal., 86, 819-827 (2007) · Zbl 1130.34033
[5] Birkhoff, G. D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., vol. IX (1927), American Mathematical Society: American Mathematical Society Providence, RI · JFM 53.0732.01
[6] Bochner, S., Curvature and Betti numbers in real and complex vector bundles, Univ. e Politec. Torino. Rend. Semin. Mat., 15, 225-253 (1955-1956) · Zbl 0072.17301
[7] Bochner, S., A new approach to almost periodicity, Proc. Natl. Acad. Sci. USA, 48, 2039-2043 (1962) · Zbl 0112.31401
[8] Bohl, P., Über die Darstellung von Funktionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten (1893), 31 pp · JFM 25.0733.01
[9] Bohl, P., Über eine Differentialgleichung der Störungstheorie, J. Reine Angew. Math., 131, 268-321 (1906), (in German) · JFM 37.0343.02
[10] Bohr, H., Sur les Fonction Presque-Periodiques, C. R. Acad. Sci. Paris, 177, 737-739 (1923) · JFM 49.0189.02
[11] Bohr, H., Zur Theorie der Fastperiodischen Funktionen mit Funktionen. I. Eine Verallgemeinerung der Theorie der Fourerreinhen, Acta. Math., 45, 29-127 (1924) · JFM 50.0196.01
[12] Bohr, H., Zur Theorie der Fastperiodischen Funktionen mit Funktionen. II. Zusammenhang der Fastperiodischen Funktionen mit Funktionen von Unendlich Vielen Variablen; Gleichmässige Approximation durch Trigonometrische Summen, Acta. Math., 46, 101-214 (1925) · JFM 51.0212.02
[13] Bohr, H., Zur Theorie der Fastperiodischen Funktionen mit Funktionen. III. Dirichletentwicklung analytischer Funktionen, Acta. Math., 47, 237-281 (1926) · JFM 52.0330.04
[14] Bohr, H., Almost Periodic Functions (1947), Chelsea Publishing Company: Chelsea Publishing Company New York, ii+114 pp
[15] Bronsteyn, I. U., Extensions of Minimal Transformation Group (1979), Stiintsa: Stiintsa Kishinev: Sijthoff & Noordhoff: Stiintsa: Stiintsa Kishinev: Sijthoff & Noordhoff Alphen Aan Den Rijn, (in Russian); English translation: Extensions of Minimal Transformation Group · Zbl 0431.54023
[16] Caraballo, T.; Cheban, D., Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition. I, J. Differ. Equ., 246, 108-128 (2009) · Zbl 1166.34021
[17] Caraballo, T.; Cheban, D., Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition. II, J. Differ. Equ., 246, 1164-1186 (2009) · Zbl 1166.34022
[18] Caraballo, T.; Cheban, D., Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations, J. Differ. Equ., 251, 708-727 (2011) · Zbl 1236.37017
[19] Caraballo, T.; Cheban, D., Almost periodic and almost automorphic solutions of linear differential equations, Discrete Contin. Dyn. Syst., 33, 1857-1882 (2013) · Zbl 1268.37007
[20] Cheban, D., Levitan almost periodic and almost automorphic solutions of V-monotone differential equations, J. Dyn. Differ. Equ., 20, 69-697 (2008) · Zbl 1151.37027
[21] Cheban, D., Asymptotically Almost Periodic Solutions of Differential Equations (2009), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York, ix+186 pp · Zbl 1222.34002
[22] Cheban, D., Global Attractors of Non-autonomous Dissipative Dynamical Systems, Interdisciplinary Mathematical Sciences, vol. 1 (2004), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ, xxiv+502 pp · Zbl 1098.37002
[23] Cheban, D.; Mammana, C., Invariant manifolds, almost periodic and almost automorphic solutions of second-order monotone equations, Int. J. Evol. Equ., 1, 319-343 (2005) · Zbl 1128.34030
[24] Cheban, D.; Schmalfuss, B., Invariant manifolds, global attractors, almost automrphic and almost periodic solutions of non-autonomous differential equations, J. Math. Anal. Appl., 340, 374-393 (2008) · Zbl 1128.37009
[25] Chen, Z.; Lin, W., Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations, J. Math. Pures Appl., 100, 476-504 (2013) · Zbl 1286.34082
[26] Da Prato, G.; Tubaro, L., Some results on periodic measures for differential stochastic equations with additive noise, Dyn. Syst. Appl., 1, 103-120 (1992) · Zbl 0758.60056
[27] Da Prato, G.; Tudor, C., Periodic and almost periodic solutions for semilinear stochastic equations, Stoch. Anal. Appl., 13, 13-33 (1995) · Zbl 0816.60062
[28] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, vol. 152 (2014), Cambridge University Press: Cambridge University Press Cambridge, xviii+493 pp · Zbl 1317.60077
[29] Daletskii, Yu. L.; Krein, M. G., Stability of Solutions of Differential Equations in Banach Space (1974), Nauka: Nauka Moscow: Amer. Math. Soc.: Nauka: Nauka Moscow: Amer. Math. Soc. Providence, RI, English transl.: · Zbl 0286.34094
[30] Feng, C.; Wu, Y.; Zhao, H., Anticipating random periodic solutions-I. SDEs with multiplicative linear noise, J. Funct. Anal., 271, 2, 365-417 (2016) · Zbl 1356.37073
[31] Fu, M.; Liu, Z., Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Am. Math. Soc., 138, 3689-3701 (2010) · Zbl 1202.60109
[32] Halanay, A., Periodic and almost periodic solutions to affine stochastic systems, (Proceedings of the Eleventh International Conference on Nonlinear Oscillations. Proceedings of the Eleventh International Conference on Nonlinear Oscillations, Budapest, 1987 (1987), János Bolyai Math. Soc.: János Bolyai Math. Soc. Budapest), 94-101 · Zbl 0627.34048
[33] Ichikawa, A., Bounded solutions and periodic solutions of a linear stochastic evolution equation, (Probability Theory and Mathematical Statistics. Probability Theory and Mathematical Statistics, Kyoto, 1986. Probability Theory and Mathematical Statistics. Probability Theory and Mathematical Statistics, Kyoto, 1986, Lecture Notes in Math., vol. 1299 (1988), Springer: Springer Berlin), 124-130 · Zbl 0633.60084
[34] Kamenskii, M.; Mellah, O.; Raynaud de Fitte, P., Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, J. Math. Anal. Appl., 427, 336-364 (2015) · Zbl 1359.60073
[35] Khasminskii, R., Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7 (1980), Sijthoff & Noordhoff: Sijthoff & Noordhoff Alphen aan den Rijn-Germantown, Md., xvi+344 pp. Translated from the Russian by D. Louvish · Zbl 0441.60060
[36] Levitan, B., Über eine Verallgemeinerung der stetigen fastperiodischen Funktionen von H. Bohr, Ann. Math. (2), 40, 805-815 (1939), (in German) · Zbl 0025.32302
[37] Levitan, B. M., Almost Periodic Functions (1953), Gosudarstv. Izdat. Tekhn-Teor. Lit.: Gosudarstv. Izdat. Tekhn-Teor. Lit. Moscow, 396 pp. (in Russian) · Zbl 1222.42002
[38] Levitan, B. M.; Zhikov, V. V., Almost Periodic Functions and Differential Equations (1982), Moscow State University Press: Moscow State University Press Moscow: Cambridge Univ. Press: Moscow State University Press: Moscow State University Press Moscow: Cambridge Univ. Press Cambridge, xi+211 pp · Zbl 0499.43005
[39] Liu, Z.; Sun, K., Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 226, 1115-1149 (2014) · Zbl 1291.60121
[40] Liu, Z.; Wang, W., Favard separation method for almost periodic stochastic differential equations, J. Differ. Equ., 260, 8109-8136 (2016) · Zbl 1335.60095
[41] Morozan, T., Periodic solutions of affine stochastic differential equations, Stoch. Anal. Appl., 4, 87-110 (1986) · Zbl 0583.60055
[42] Sell, G. R., Lectures on Topological Dynamics and Differential Equations, Van Nostrand Reinhold Math. Studies, vol. 2 (1971), Van Nostrand-Reinbold: Van Nostrand-Reinbold London · Zbl 0212.29202
[43] Shcherbakov, B. A., A certain class of Poisson stable solutions of differential equations, Differ. Uravn., 4, 2, 238-243 (1968), (in Russian) · Zbl 0182.42401
[44] Shcherbakov, B. A., Topologic Dynamics and Poisson Stability of Solutions of Differential Equations (1972), Ştiinţa: Ştiinţa Chişinău, 231 pp. (in Russian) · Zbl 0324.34042
[45] Shcherbakov, B. A., The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differ. Uravn.. Differ. Uravn., Differ. Equ., 11, 7, 937-943 (1975), (in Russian); English translation: · Zbl 0337.34040
[46] Shcherbakov, B. A., Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations (1985), Ştiinţa: Ştiinţa Chişinău, 147 pp. (in Russian) · Zbl 0638.34046
[47] Shcherbakov, B. A.; Cheban, D., Asymptotically Poisson stable motions of dynamical systems and comparability of their reccurence in limit, Differ. Uravn.. Differ. Uravn., Differ. Equ., 13, 5, 618-624 (1978), (in Russian); English translation: · Zbl 0396.34048
[48] Shcherbakov, B. A.; Fal’ko, N. S., The minimality of sets and the Poisson stability of motions in homomorphic dynamical systems, Differ. Uravn.. Differ. Uravn., Differ. Equ., 13, 6, 755-758 (1978), (in Russian); English translation: · Zbl 0396.34039
[49] Shen, W.; Yi, Y., Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Am. Math. Soc., 136, 647 (1998), x+93 pp · Zbl 0913.58051
[50] Sibirsky, K. S., Introduction to Topological Dynamics (1975), RIA AN MSSR: RIA AN MSSR Kishinev: Noordhoff: RIA AN MSSR: RIA AN MSSR Kishinev: Noordhoff Leyden, (in Russian); English translation: Introduction to Topological Dynamics · Zbl 0297.54001
[51] Tudor, C., Almost periodic solutions of affine stochastic evolution equations, Stoch. Stoch. Rep., 38, 251-266 (1992) · Zbl 0752.60049
[52] Wang, Y.; Liu, Z., Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25, 2803-2821 (2012) · Zbl 1260.60114
[53] Veech, W. A., Almost automorphic functions on groups, Am. J. Math., 87, 719-751 (1965) · Zbl 0137.05803
[54] Xia, Z., Pseudo almost automorphic in distribution solutions of semilinear stochastic integro-differential equations by measure theory, Int. J. Math., 26, 13, Article 1550112 pp. (2015) · Zbl 1333.60128
[55] Zhao, H.; Zheng, Z., Random periodic solutions of random dynamical systems, J. Differ. Equ., 246, 5, 2020-2038 (2009) · Zbl 1162.37023
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