We study stochastic differential equations with simple additive noise under the impact of a shear force. We fully characterize the behaviour of these systems, depending on the shear function, in terms of global and local properties of the stochastic flow.
In more detail, we show criteria for weak and strong completeness and the existence of random set and point attractors. We also demonstrate the possibility of negative and positive Lyapunov exponents, in combination with all other behaviours, indicating synchronization or chaos as characterizations of the random attractors if they exist.