On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusio n vector fields is finite dimensional and solvable, then the Row is conjuga te to the Row of a non-autonomous random differential equation. i.e. one ca n be transformed into the other via a random diffeomorphism of d-dimensiona l Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors i n terms of the random differential equation. which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise .