The motion of monatomic steps on Si(001) is studied on an atomic scale at elevated temperatures with scanning tunneling microscopy. The kinks in the step edges move in units of two dimers along the monatomic A-type step edge and perpendicular to the monatomic B-type step edge. The overall time dependencies of the equilibrium step fluctuations of A- and B-type step edges were found to be both proportional to t0.6±0.1. The fluctuations of long kinks in the B-type step edge are, however, much larger and exhibit initially a linear t dependence, i.e., one-dimensional random-walk behavior. Both time dependencies can be understood in terms of the Langevin equation.