Stratosphere-troposphere exchange (STE) and its effects on the stratospheric and tropospheric chemical compositions have been studied for the past two decades, but details on how mass is transported between the stratosphere and the troposphere are not well established. The goal of this study is to better describe global properties of cross tropopause trajectories, and to understand the processes related to transport of mass between the troposphere and the stratosphere. This understanding led us to build the simplest model which captures the most important properties of STE. To do this, nine-day extra-tropical stratosphere-troposphere exchange trajectories covering a period of 10 years, calculated using the ERA-15 re-analysis data, are investigated. The present study shows that the fraction of trajectories that reside in the stratosphere or in the troposphere does not depend on the direction of the exchange (stratosphere-to-troposphere transport, STT, or troposphere-to-stratosphere transport, TST). Trajectories are found to reside longer in the troposphere than in the stratosphere which suggests that they are driven down by asymmetric two-way motion. A random walk model is used to see whether this asymmetric transport is a result of a diffusive process. The transport of trajectories along isentropic coordinates is found to be compatible with a Brownian motion with higher probabilities to go downward. Since stratosphere-troposphere exchange reflects a differential motion of air masses and the tropopause, the potential temperature at the tropopause directly above or below the air mass is also investigated. The tropopause steps distributions are not stationary and they show some dynamical behaviors like the deformation of the tropopause at exchange time. Dispersion of trajectories in the atmosphere was furthermore investigated using several methods. They gave rise to three different transport mechanisms: diffusion, sub-diffusion and super-diffusion transports.