The conjugacy of stochastic and random differential equations and the existence of global attractors
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چکیده
We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f0 and the diffusion vector fields f1, · · · , fm, and investigate the existence of global random attractors for the associated flows φ. For this purpose φ is decomposed into a stationary diffeomorphism Φ given by the stochastic differential equation on the space of smooth flows on R driven by m independent stationary Ornstein Uhlenbeck processes z1, · · · , z and the vector fields f1, · · · , fm, and a flow χ generated by the non-autonomous ordinary differential equation given by the vector field (t ∂x )[f0(Φt) + ∑m i=1 fi(Φt) z i t]. In this setting, attractors of χ are canonically related with attractors of φ. For χ, the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f0 is still a Lyapunov function for χ, yielding an attractor this way. The criterion is finally tested in various prominent examples.
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تاریخ انتشار 1998