Numerical attractors for rough differential equations
Hoang Duc Luu and Peter Kloeden
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Submission date: 25. Oct. 2021 (revised version: June 2023)
Keywords and phrases: rough differential equations, Euler scheme, random dynamical systems, random attractors
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We study the explicit Euler scheme to approximate the solutions of rough differential equations under a bounded or linear diffusion term, where the drift term satisfies a local Lipschitz continuity and a bounded linear growth condition. The Euler scheme is then proved to converge for a given solution, although the approximation of the error depends on the initial condition. For a dissipative drift term with linear growth condition and a bounded diffusion term, the numerical solution under a regular grid generates a random dynamical system which admits a random pullback attractor. We also prove that for bounded drift and diffusion terms, the numerical pullback attractor then converges upper semi-continuously to the continuous-time pullback attractor as the time step goes to zero.