We consider a process that starts at height y, stays there for a time X0 ∼ exp(y) when it drops to a level Z1 ∼ U(0, y). Thereafter it stays at level Zn for time Xn ∼ exp(Zn), then drops to a level Zn+1 ∼ U(0,Zn). We investigate properties of this process, as well as the Poisson hyperbolic process which is obtained by randomizing the starting point y of the above process. This process is associated with a rate 1 Poisson process in the positive quadrant: Its path is the minimal RCLL decreasing step function through Poisson points in the positive quadrant. The finite dimensional distributions are then multivariate exponential in sense of Marshall-Olkin.