# American Institute of Mathematical Sciences

November  2019, 24(11): 5927-5944. doi: 10.3934/dcdsb.2019113

## Almost periodic solutions and stable solutions for stochastic differential equations

 1 School of Mathematics, Jilin University, Changchun 130012, China 2 Center for Mathematics and Interdisciplinary Sciences, and School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China 3 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 4 School of Basic Sciences, Changchun University of Technology, Changchun 130012, China

* Corresponding author

Received  September 2018 Published  June 2019

Fund Project: The first author is supported by National Research Program of China Grant 2013CB834100, Jilin DRC Grant 2017C028-1, NSFC Grants 11571065 and 11171132; the second author is supported by NSFC Grants 11522104 and 11871132, and the startup and Xinghai Jieqing funds from Dalian University of Technology

In this paper, we discuss the relationships between stability and almost periodicity for solutions of stochastic differential equations. Our essential idea is to get stability of solutions or systems by some inherited properties of Lyapunov functions. Under suitable conditions besides Lyapunov functions, we obtain the existence of almost periodic solutions in distribution.

Citation: Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
##### References:
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Funktionen einer Variablen, (German), Math. Ann., 96 (1927), 119-147. doi: 10.1007/BF01209156. Google Scholar [7] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043. doi: 10.1073/pnas.48.12.2039. Google Scholar [8] H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅰ., Acta Math., 45 (1925), 29-127. doi: 10.1007/BF02395468. Google Scholar [9] H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅱ., Acta Math., 46 (1925), 101-214. doi: 10.1007/BF02543859. Google Scholar [10] H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅲ., Acta Math., 47 (1926), 237-281. doi: 10.1007/BF02543846. Google Scholar [11] K. L. Chung, A Course in Probability Theory, 3nd edition, Academic press, Inc., San Diego, CA, 2001. xviii+419 pp. Google Scholar [12] W. A. Coppel, Almost periodic properties of ordinary differential equations, Ann. Mat. Pura Appl.(4), 76 (1967), 27-49. doi: 10.1007/BF02412227. Google Scholar [13] G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33. doi: 10.1080/07362999508809380. Google Scholar [14] L. G. Deysach and G. R. Sell, On the existence of almost periodic motion, Michgan Math. J., 12 (1965), 87-95. doi: 10.1307/mmj/1028999248. Google Scholar [15] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377. Springer-Verlag, Berlin-New York, 1974. viii+336 pp. Google Scholar [16] A. Halanay, Periodic and almost periodic solutions to affine stochastic systems, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations, (Budapest, 1987), 94–101, János Bolyai Math. Soc., Budapest, 1987. Google Scholar [17] J. Hale, Periodic and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 15 (1964), 289-304. doi: 10.1007/BF00249199. Google Scholar [18] I. Ya. Kac and N. N. Krasovskii, On the stability of systems with random parameters, (Russian), J. Appl. Math. Mech., 24 (1960), 1225-1246. Google Scholar [19] R. Khasminskii, On the stability of the trajectory of Markov processes,, J. Appl. Math. Mech., 26 (1962), 1554-1565. doi: 10.1016/0021-8928(62)90192-2. Google Scholar [20] R. Khasminskii, Stochastic Stability of Differential Equations, 1$^{nd}$ edition, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980. xvi+344 pp; 2$^{nd}$ edition, Springer, Heidelberg, 2012. xviii+339 pp. Google Scholar [21] F. Kozin, On almost sure stability of linear systems with random coefficients, J. Math. Phys., 42 (1963), 59-67. doi: 10.1002/sapm196342159. Google Scholar [22] H. J. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, Vol. 33 Academic Press, New York-London, 1967. xiv+161 pp. Google Scholar [23] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Translated from the Russian by L. W. Longdon, Cambridge University Press, CambridgeNew York, 1982. xi+211 pp. Google Scholar [24] Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136. doi: 10.1016/j.jde.2016.02.019. Google Scholar [25] X. Mao, Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994. xii+307 pp. Google Scholar [26] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics and Applications, Horwood Publishing Limited, Chichester, 1997. xii+366 pp. Google Scholar [27] A. A. Markov, Stabilität in Liapvanoffschen Sinne und Fastperiodizität, Math. Z., 36 (1933), 708-738. doi: 10.1007/BF01188645. Google Scholar [28] R. K. Miller, Almost periodic differential equations as dynamical systems with applications to the extence of a. p. solution, J. Differential Equations, 1 (1965), 337-345. doi: 10.1016/0022-0396(65)90012-4. Google Scholar [29] T. Morozan and C. Tudor, Almost periodic solutions of affine Itô equations, Stoch. Anal. Appl., 7 (1989), 451-474. doi: 10.1080/07362998908809194. Google Scholar [30] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London, 1967. xi+276 pp. Google Scholar [31] G. Seifert, Almost periodic solutions for almost periodic systems of ordinary differential equations, J. Differential Equations, 2 (1966), 305-319. doi: 10.1016/0022-0396(66)90071-4. Google Scholar [32] G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262. doi: 10.2307/1994645. Google Scholar [33] G. R. Sell, Nonautonomous differential equations and topological dynamics. Ⅱ. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283. doi: 10.1090/S0002-9947-1967-0212314-4. Google Scholar [34] C. Vârsana, Asymptotic almost periodic solutions for stochastic differential equations, Tôhoku Math. J., 41 (1989), 609-618. doi: 10.2748/tmj/1178227731. Google Scholar [35] T. Yoshizawa, Extreme stability and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 17 (1964), 148-170. doi: 10.1007/BF00253052. Google Scholar [36] T. Yoshizawa, Asymptotically almost periodic solutions of an almost periodic system, Funkcial. Ekvac., 12 (1969), 23-40. Google Scholar [37] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975. vii+233 pp. Google Scholar

show all references

##### References:
 [1] L. Amerio and G. Prouse, Almost-periodic Functions and Functional Equations, Van Nostrand Reinhold Co., New York-Toronto, Ont.-Melbourne, 1971. viii+184 pp. Google Scholar [2] L. Arnold and C. Tudor, Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics Stochastics Rep., 64 (1998), 177-193. doi: 10.1080/17442509808834163. Google Scholar [3] G. K. Basak, A class of limit theorems for singular diffusions, J. Multivariate Anal., 39 (1991), 44-59. doi: 10.1016/0047-259X(91)90004-L. Google Scholar [4] G. K. Basak and R. N. Bhattachaya, Stability in distribution for a class of singular diffusions, Ann. Probab., 20 (1992), 312-321. doi: 10.1214/aop/1176989928. Google Scholar [5] J. E. Bertram and P. E. Sarachik, Stability of circuits with randomly time-varying parameters, IRE Trans. Information Theory, 6 (1959), 260-270. Google Scholar [6] S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, I. Funktionen einer Variablen, (German), Math. Ann., 96 (1927), 119-147. doi: 10.1007/BF01209156. Google Scholar [7] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043. doi: 10.1073/pnas.48.12.2039. Google Scholar [8] H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅰ., Acta Math., 45 (1925), 29-127. doi: 10.1007/BF02395468. Google Scholar [9] H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅱ., Acta Math., 46 (1925), 101-214. doi: 10.1007/BF02543859. Google Scholar [10] H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅲ., Acta Math., 47 (1926), 237-281. doi: 10.1007/BF02543846. Google Scholar [11] K. L. Chung, A Course in Probability Theory, 3nd edition, Academic press, Inc., San Diego, CA, 2001. xviii+419 pp. Google Scholar [12] W. A. Coppel, Almost periodic properties of ordinary differential equations, Ann. Mat. Pura Appl.(4), 76 (1967), 27-49. doi: 10.1007/BF02412227. Google Scholar [13] G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33. doi: 10.1080/07362999508809380. Google Scholar [14] L. G. Deysach and G. R. Sell, On the existence of almost periodic motion, Michgan Math. J., 12 (1965), 87-95. doi: 10.1307/mmj/1028999248. Google Scholar [15] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377. Springer-Verlag, Berlin-New York, 1974. viii+336 pp. Google Scholar [16] A. Halanay, Periodic and almost periodic solutions to affine stochastic systems, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations, (Budapest, 1987), 94–101, János Bolyai Math. Soc., Budapest, 1987. Google Scholar [17] J. Hale, Periodic and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 15 (1964), 289-304. doi: 10.1007/BF00249199. Google Scholar [18] I. Ya. Kac and N. N. Krasovskii, On the stability of systems with random parameters, (Russian), J. Appl. Math. Mech., 24 (1960), 1225-1246. Google Scholar [19] R. Khasminskii, On the stability of the trajectory of Markov processes,, J. Appl. Math. Mech., 26 (1962), 1554-1565. doi: 10.1016/0021-8928(62)90192-2. Google Scholar [20] R. Khasminskii, Stochastic Stability of Differential Equations, 1$^{nd}$ edition, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980. xvi+344 pp; 2$^{nd}$ edition, Springer, Heidelberg, 2012. xviii+339 pp. Google Scholar [21] F. Kozin, On almost sure stability of linear systems with random coefficients, J. Math. Phys., 42 (1963), 59-67. doi: 10.1002/sapm196342159. Google Scholar [22] H. J. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, Vol. 33 Academic Press, New York-London, 1967. xiv+161 pp. Google Scholar [23] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Translated from the Russian by L. W. Longdon, Cambridge University Press, CambridgeNew York, 1982. xi+211 pp. Google Scholar [24] Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136. doi: 10.1016/j.jde.2016.02.019. Google Scholar [25] X. Mao, Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994. xii+307 pp. Google Scholar [26] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics and Applications, Horwood Publishing Limited, Chichester, 1997. xii+366 pp. Google Scholar [27] A. A. Markov, Stabilität in Liapvanoffschen Sinne und Fastperiodizität, Math. Z., 36 (1933), 708-738. doi: 10.1007/BF01188645. Google Scholar [28] R. K. Miller, Almost periodic differential equations as dynamical systems with applications to the extence of a. p. solution, J. Differential Equations, 1 (1965), 337-345. doi: 10.1016/0022-0396(65)90012-4. Google Scholar [29] T. Morozan and C. Tudor, Almost periodic solutions of affine Itô equations, Stoch. Anal. Appl., 7 (1989), 451-474. doi: 10.1080/07362998908809194. Google Scholar [30] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London, 1967. xi+276 pp. Google Scholar [31] G. Seifert, Almost periodic solutions for almost periodic systems of ordinary differential equations, J. Differential Equations, 2 (1966), 305-319. doi: 10.1016/0022-0396(66)90071-4. Google Scholar [32] G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262. doi: 10.2307/1994645. Google Scholar [33] G. R. Sell, Nonautonomous differential equations and topological dynamics. Ⅱ. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283. doi: 10.1090/S0002-9947-1967-0212314-4. Google Scholar [34] C. Vârsana, Asymptotic almost periodic solutions for stochastic differential equations, Tôhoku Math. J., 41 (1989), 609-618. doi: 10.2748/tmj/1178227731. Google Scholar [35] T. Yoshizawa, Extreme stability and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 17 (1964), 148-170. doi: 10.1007/BF00253052. Google Scholar [36] T. Yoshizawa, Asymptotically almost periodic solutions of an almost periodic system, Funkcial. Ekvac., 12 (1969), 23-40. Google Scholar [37] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975. vii+233 pp. Google Scholar
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