Geodesic
The comparability of motions in dynamical systems and recurrent solutions of (S)PDEs
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 53-83.

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Shcherbakov's comparability method is very useful to study recurrent solutions of differential equations. In this paper, we extend the method from metric spaces to uniform spaces, which applies well to dynamical systems in infinite-dimensional spaces. This generalized comparability method can be easily used to study recurrent solutions of (stochastic) partial differential equations under weaker conditions than in earlier results. We also show that the distribution of solutions of SDEs naturally generates a semiflow or skew-product semiflow on the space of probability measures, which is interesting in itself. As illustration, we give an application to semilinear stochastic partial differential equations. 
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David Cheban; Zhenxin Liu. The comparability of motions in dynamical systems and recurrent solutions of (S)PDEs. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 53-83. https://geodesic-test.mathdoc.fr/item/BASM_2024_1_a4/

[1] P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication, Second edition, John Wiley Sons, Inc., New York, 1999, x+277 pp. | MR

[2] S. Bochner, “A new approach to almost periodicity”, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039–2043 | DOI | MR | Zbl

[3] Extensions of Minimal Transformation Group, Sijthoff Noordhoff, Alphen aan den Rijn, 1979

[4] T. Caraballo and D. Cheban, “Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I”, J. Differential Equations, 246 (2009), 108–128 | DOI | MR | Zbl

[5] T. Caraballo and D. Cheban, “Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. II”, J. Differential Equations, 246 (2009), 1164–1186 | DOI | MR | Zbl

[6] T. Caraballo and D. Cheban, “Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations”, J. Differential Equations, 251 (2011), 708–727 | DOI | MR | Zbl

[7] T. Caraballo and D. Cheban, “Almost periodic and almost automorphic solutions of linear differential equations”, Discrete Contin. Dyn. Syst., 33 (2013), 1857–882 | DOI | MR

[8] D. Cheban, “Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations”, J. Dynam. Differential Equations, 20 (2008), 69–697 | DOI | MR

[9] D. Cheban, Global Attractors of Non-autonomous Dynamical and Control Systems, Interdisciplinary Mathematical Sciences, 18, Second edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015, xxvi+589 pp. | DOI | MR | Zbl

[10] D. Cheban and Z. Liu, “Poisson stable motions of monotone nonautonomous dynamical systems”, Sci. China Math., 62 (2019), 1391–1418 | DOI | MR | Zbl

[11] 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 | MR | Zbl

[12] D. Cheban and Z. Liu, “Averaging principle on infinite intervals for stochastic ordinary differential equations”, Electron. Res. Arch., 29 (2021), 2791–2817 | DOI | MR | Zbl

[13] D. Cheban and C. Mammana, “Invariant manifolds, almost periodic and almost automorphic solutions of second-order monotone equations”, Int. J. Evol. Equ., 1 (2005), 319–343 | MR | Zbl

[14] D. Cheban and B. Schmalfuss, “Invariant manifolds, global attractors, almost automrphic and almost periodic solutions of non-autonomous differential equations”, J. Math. Anal. Appl., 340 (2008), 374–393 | DOI | MR | Zbl

[15] M. Cheng and Z. Liu, “The second Bogolyubov theorem and global averaging principle for SPDEs with monotone coefficients”, SIAM J. Math. Anal., 55 (2023), 1100–1144 | DOI | MR | Zbl

[16] M. Cheng, Z. Liu, and M. Röckner, “Averaging principle for stochastic complex Ginzburg-Landau equations”, J. Differential Equations, 368 (2023), 58–104 | DOI | MR | Zbl

[17] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992, xviii+454 pp. | MR | Zbl

[18] R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2002, x+555 pp. | MR | Zbl

[19] I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Translated from Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, 72, Springer-Verlag, New York-Heidelberg, 1972, viii+354 pp. | MR | Zbl

[20] J. L. Kelley, General Topology, Graduate Texts in Mathematics, 27, Springer-Verlag, New York-Berlin, 1975, xiv+298 pp. ; Reprint:, Van Nostrand, Toronto, Ont., 1955 | MR | Zbl | Zbl

[21] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Translated from Russian by L. W. Longdon, Cambridge University Press, Cambridge-New York, 1982, xi+211 pp. | MR | Zbl

[22] X. Liu and Z. Liu, “Poisson stable solutions for stochastic differential equations with Lévy noise”, Acta Math. Sin. (Engl. Ser.), 38 (2022), 22–54 | DOI | MR | Zbl

[23] H. Poincaré, Methodes nouvelles de la mécanique célèste, v. 1, Gauthier-Villars, Paris, 1892 (in French) | MR

[24] G. R. Sell, Lectures on Topological Dynamics and Differential Equations, Van Nostrand Reinhold math. studies, 2, Van Nostrand–Reinbold, London, 1971 | MR

[25] B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972, 231 pp. (in Russian) | MR

[26] Differ. Equ., 11:7 (1975), 937–943 | MR | Zbl

[27] B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Ştiinţa, Chişinău, 1985, 147 pp. (in Russian) | MR

[28] Differential Equations, 13:5 (1978), 618–624 | MR | Zbl

[29] Differential Equations, 13:6 (1978), 755–758 | MR | Zbl

[30] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136, no. 647, 1998, x+93 pp. | MR

[31] K. S. Sibirsky, Introduction to Topological Dynamics, Translated from Russian by Leo F. Boron, Noordhoff International Publishing, Leiden, 1975, ix+163 pp. | MR | Zbl