  Space-time correlation functions constitute a useful instrument from the
research toolkit of continuous-media and many-body physics. We adopt here this
concept for single-particle random walks and demonstrate that the corresponding
space-time velocity auto-correlation functions reveal correlations which extend
in time much longer than estimated with the commonly employed temporal
correlation functions. A generic feature of considered random-walk processes is
an effect of velocity echo identified by the existence of time-dependent
regions where most of the walkers are moving in the direction opposite to their
initial motion. We discuss the relevance of the space-time velocity correlation
functions for the experimental studies of cold atom dynamics in an optical
potential and charge transport on micro- and nano-scales.
