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,...,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 Rd driven by m independent stationary Ornstein Uhlenbeck processes z1,...,zm and the vector fields f1,...,fm, and a flow χ generated by the nonautonomous ordinary differential equation given by the vector field (∂Φt/∂x)−1[f0(Φt)+∑i=11fi(Φt)zti]. 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.