Averaging principle on infinite intervals for stochastic ordinary differential equations with Lévy noise

Xin Liu; Yan Wang; Xin Liu; Yan Wang

doi:10.3934/math.2021314

AIMS Mathematics

2021, Volume 6, Issue 5: 5316-5350. doi: 10.3934/math.2021314

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Research article

Averaging principle on infinite intervals for stochastic ordinary differential equations with Lévy noise

Xin Liu ,
Yan Wang ^,

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received: 16 January 2021 Accepted: 03 March 2021 Published: 12 March 2021
MSC : 34C29, 60H10, 60G51, 37B20, 34C27

In this paper, we establish an averaging principle on the infinite time intervals for semilinear stochastic ordinary differential equations with Lévy noise. In particular, under suitable conditions we prove that if the coefficients are Poisson stable (including periodic, quasi-periodic, almost periodic, almost automorphic etc), then there exists a unique $ \mathcal L^2 $-bounded solution of the original equation, which inherits the recurrence property of the coefficients, and the recurrent solution uniformly converges to the stationary solution of the averaged equation on the whole real axis in distribution sense.
- averaging principle,
- stochastic differential equations,
- Lévy noise,
- periodic solution,
- quasi-periodic solution,
- almost periodic solution,
- Poisson stable solution
Citation: Xin Liu, Yan Wang. Averaging principle on infinite intervals for stochastic ordinary differential equations with Lévy noise[J]. AIMS Mathematics, 2021, 6(5): 5316-5350. doi: 10.3934/math.2021314

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Abstract

In this paper, we establish an averaging principle on the infinite time intervals for semilinear stochastic ordinary differential equations with Lévy noise. In particular, under suitable conditions we prove that if the coefficients are Poisson stable (including periodic, quasi-periodic, almost periodic, almost automorphic etc), then there exists a unique $ \mathcal L^2 $-bounded solution of the original equation, which inherits the recurrence property of the coefficients, and the recurrent solution uniformly converges to the stationary solution of the averaged equation on the whole real axis in distribution sense.

References

[1]	N. N. Bogolyubov, N. M. Krylov, La theorie generale de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire, Ann. Math., 38 (1937), 65-113.
[2]	N. M. Krylov, N. N. Bogolyubov, Introduction to Non-Linear Mechanics, Princeton, N. J.: Princeton University Press, 1943.
[3]	N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics, Kiev: Akademiya Nauk Ukrainskoi SSR, 1945.
[4]	N. N. Bogolyubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, New York: Gordon and Breach Science Publishers, 1961.
[5]	R. L. Stratonovich, Topics in the Theory of Random Noise, New York: Gordon and Breach, 1963.
[6]	S. Cerrai, M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Relat. Fields, 144 (2009), 137-177. doi: 10.1007/s00440-008-0144-z
[7]	S. Cerrai, 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
[8]	J. Duan, W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Amsterdam: Elsevier, 2014.
[9]	M. Freidlin, A. Wentzell, Random Perturbations of Dynamical Systems, Heidelberg: Springer, 2012.
[10]	R. Z. Khasminskii, On the principle of averaging the Itô's stochastic differential equations, Kybernetika, 4 (1968), 260-279.
[11]	A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Providence, R. I.: American Mathematical Society, 1989.
[12]	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
[13]	I. Vrkoc, Weak averaging of stochastic evolution equations, Math. Bohem., 120 (1995), 91-111. doi: 10.21136/MB.1995.125891
[14]	W. Wang, A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differ. Equations, 253 (2012), 1265-1286. doi: 10.1016/j.jde.2012.05.011
[15]	Y. Xu, J. Duan, W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Phys. D, 240 (2011), 1395-1401. doi: 10.1016/j.physd.2011.06.001
[16]	W. Mao, S. You, X. Wu, X. Mao, On the averaging principle for stochastic delay differential equations with jumps, Adv. Differ. Equations, 70 (2015), 1-19.
[17]	V. Burd, Method of Averaging for Differential Equations on an Infinite Interval: Theory and applications, New York: Chapman and Hall, 2007.
[18]	D. Cheban, Z. Liu, Averaging principle on infinite intervals for stochastic ordinary differential equations, Electron. Res. Arch., 2021. Available from: http://www.aimsciences.org/article/doi/10.3934/era.2021014.
[19]	D. Cheban, Z. Liu, Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations, J. Differ. Equations, 269 (2020), 3652-3685.
[20]	M. Cheng, Z. Liu, Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients, 2021. Available from: https://arXiv.org/abs/1911.02169.
[21]	X. Liu, Z. Liu, Poisson stable solutions for stochastic differential equations with Lévy noise, 2021. Available from: https://arXiv.org/abs/2002.00395.
[22]	G. R. Sell, Topological Dynamics and Ordinary Differential Equations, London: Van Nostrand Reinhold Co., 1971.
[23]	B. A. Shcherbakov, Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Chişinǎu: Ştiinţa, 1972.
[24]	B. A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Chişinǎu: Ştiinţa, 1985.
[25]	K. S. Sibirsky, Introduction to Topological Dynamics, Leiden: Noordhoff International Publishing, 1975.
[26]	D. Cheban, Global Attractors of Nonautonomous Dynamical and Control Systems, 2Eds., Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2015.
[27]	B. M. Levitan, V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow: Moscow State University Press, 1978.
[28]	B. A. Shcherbakov, A certain class of Poisson stable solutions of differential equations, Differencial'nye Uravnenija, 4 (1968), 238-243.
[29]	B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differentcial'nye Uravnenija, 11 (1975), 1246-1255.
[30]	D. Applebaum, Lévy Process and Stochastic Calculus, 2Eds., Cambridge: Cambridge University Press, 2009.
[31]	H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, In: M. M. Rao, Real and Stochastic Analysis, Boston: Birkhäuser, (2004), 305-373.
[32]	G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2Eds., Cambridge: Cambridge University Press, 2014.
[33]	Ju. L. Daleckii, M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Providence, R. I.: American Mathematical Society, 1974.
[34]	D. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, New York: Hindawi Publishing Corporation, 2009.
[35]	R. M. Dudley, Real Analysis and Probability, 2Eds., Cambridge: Cambridge University Press, 2002.
[36]	M. A. Krasnoselskii, V. Burd, Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations, Moscow: Nauka, 1970.

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