The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

Abstract

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,...,f_m, 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^d driven by m independent stationary Ornstein Uhlenbeck processes z¹,...,z^m and the vector fields f₁,...,f_m, and a flow χ generated by the nonautonomous ordinary differential equation given by the vector field (∂Φ_t/∂x)⁻¹[f₀(Φ_t)+∑ ¹_i=1 f_i(Φ_t)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.