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Random attractors for stochastic differential equations driven by two-sided Lévy processes. (English) Zbl 1428.37050
Summary: In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.

##### MSC:
 37G35 Dynamical aspects of attractors and their bifurcations 37H20 Bifurcation theory for random and stochastic dynamical systems 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 37H10 Generation, random and stochastic difference and differential equations 60G65 Nonlinear processes (e.g., $$G$$-Brownian motion, $$G$$-Lévy processes)
##### Keywords:
Lévy process; random dynamical system; random attractor
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##### References:
 [1] Schmalfuß, B.; Reitmann, V.; Riedrich, T.; Koksch, N., Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, 185-192, (1992), Dresden: Technische Universität, Dresden [2] Crauel, H.; Flandoli, F., Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100, 3, 365-393, (1994) · Zbl 0819.58023 [3] Flandoli, F.; Schmalfuß, B., Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastic Rep., 59, 1-2, 21-45, (1996) · Zbl 0870.60057 [4] Schoutens, W., Lévy Processes in Finance: Pricing Financial Derivatives, (2003), Chichester, UK: Wiley, Chichester, UK [5] Xu, Y.; Li, H.; Wang, H.; Jia, W.; Yue, X.; Kurths, J., The estimates of the mean first exit time of a bistable system excited by Poisson white noise, J. Appl. Mech. Mech., 84, 9, 091004, (2017) [6] Xu, Y.; Wu, J.; Du, L.; Yang, H., Stochastic resonance in a genetic toggle model with harmonic excitation and Lévy noise, Chaos Solitons Fract, 92, 91-100, (2016) · Zbl 1372.92067 [7] Xu, Y.; Pei, B.; Li, Y., Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci, 38, 11, 2120-2131, (2015) · Zbl 1345.60051 [8] Pei, B.; Xu, Y.; Wu, J., Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles, J. Math. Anal. Appl, 447, 1, 243-268, (2017) · Zbl 1387.60102 [9] Kümmel, K., On the dynamics of Marcus type stochastic differential equations, (2016), Friedrich-Schiller-Universität: Friedrich-Schiller-Universität, Jena [10] Xu, Y.; Pei, B.; Guo, G., Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise, Appl. Math. Comput., 263, 398-409, (2015) · Zbl 1410.60060 [11] Pei, B.; Xu, Y., Mild solutions of local non-Lipschitz stochastic evolution equations with jumps, Appl. Math. Lett., 52, 80-86, (2016) · Zbl 1356.60106 [12] Liu, X., On asymptotic phenomena of dynamical systems driven by Lévy processes. Ph.D, (2010), Huazhong University of Science & Technology: Huazhong University of Science & Technology, China [13] Liu, X.; Duan, J.; Liu, J.; Kloeden, P. E., Synchronization of systems of Marcus canonical equations driven by α-stable noises, Nonlinear Anal.: Real World Appl., 11, 5, 3437-3445, (2010) · Zbl 1203.37124 [14] Liu, X.; Duan, J.; Liu, J.; Kloeden, P. E., Synchronization of dissipative dynamical systems driven by non-Gaussian Lévy noises, Int. J. Stochastic Anal., 2010, 1, (2010) · Zbl 1218.60051 [15] Cornfeld, I. P.; Fomin, S. V.; Sinai, Y. G., Ergodic Theory, (1982), Berlin: Springer, Berlin [16] Marcus, S., Modeling and analysis of stochastic differential equations driven by point processes, IEEE Trans. Inform. Theory, 24, 2, 164-172, (1978) · Zbl 0372.60084 [17] Marcus, S., Modeling and approximation of stochastic differential equations driven by semimartingales, Stochastics, 4, 3, 223-245, (1981) · Zbl 0456.60064 [18] Kurtz, T.; Pardoux, E.; Protter, P., Stratonovich stochastic differential equations driven by general semimartingales, Ann. de l’IHP Probab. Stat, 31, 351-377, (1995) · Zbl 0823.60046 [19] Sekimoto, K., Stochastic Energetics, (2010), Berlin: Springer, Berlin [20] Arnold, L.; Schmalfuß, B.; Naess, A.; Krenk, S., IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, Fixed points and attractors for random dynamical systems, 19-28, (1996), Dordrecht: Springer, Dordrecht [21] Applebaum, D., Lévy Processes and Stochastic Calculus, (2009), Cambridge, UK: Cambridge University Press, Cambridge, UK [22] Arnold, L., Random Dynamical Systems, (1998), Berlin: Springer, Berlin [23] Billingsley, P., Convergence of Probability Measures, (1999), New York: John Wiley & Sons, New York · Zbl 0172.21201 [24] Lachout, P., A Skorohod space of discontinuous functions on a general set, Acta Univ. Carolinae. Math. Phys., 33, 2, 91-97, (1992) · Zbl 0787.60043 [25] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis: A Hitchhiker’s Guide, (2007), Berlin: Springer, Berlin · Zbl 0938.46001 [26] Liptser, R.; Shiryaev, A., Statistics of Random Processes: I. General Theory, (2013), Berlin: Springer Science & Business Media, Berlin [27] Protter, P., Stochastic Differential Equations. Stochastic Integration and Differential Equations, 249-361, (2005), New York: Springer, New York [28] Rao, M.; Kunita, H., Real and Stochastic Analysis: New Perspectives - Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphism, (2004), Boston: Birkhäuser, Boston [29] Jurek, Z.; Vervaat, W., An integral representation for selfdecomposable Banach space valued random variables, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 62, 2, 247-262, (1983) · Zbl 0488.60028 [30] Duan, J.; Lu, K.; Schmalfuß, B., Invariant manifolds for stochastic partial differential equations, Ann. Probab, 31, 4, 2109-2135, (2003) · Zbl 1052.60048 [31] Coddington, E.; Levinson, N., Theory of Ordinary Differential Equations, (1955), New Delhi: Tata McGraw-Hill Education, New Delhi · Zbl 0064.33002
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