  We investigate the stability properties of discrete and hybrid stochastic
nonlinear dynamical systems. More precisely, we extend the stochastic
contraction theorems (which were formulated for continuous systems) to the case
of discrete and hybrid resetting systems. In particular, we show that the mean
square distance between any two trajectories of a discrete (or hybrid
resetting) contracting stochastic system is upper-bounded by a constant after
exponential transients. Using these results, we study the synchronization of
noisy nonlinear oscillators coupled by discrete noisy interactions.
