  We analyze common lifts of stochastic processes to rough paths/rough
drivers-valued processes and give sufficient conditions for the cocycle
property to hold for these lifts. We show that random rough differential
equations driven by such lifts induce random dynamical systems. In particular,
our results imply that rough differential equations driven by the lift of
fractional Brownian motion in the sense of Friz-Victoir induce random dynamical
systems.
