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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)
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