Drift and diffusion in a periodic two-dimensional velocity field without attractors.

We consider the transport of overdamped particles in a two-dimensional periodic velocity field. This field possesses extended lines of fixed points where the deterministic motion stops. Additive noise makes the lines penetrable and results in an oscillatory motion along tori. We characterize the stochastic motion by the probability distribution density, the stationary mean velocity, and the mean times of escape from bounded domains. For intermediate noise intensities, the fluctuations enhance the transport of the particles compared to the deterministic case. A fast dichotomic modulation of asymmetry enhances fluxes.