  A representation of the continuum fermionic propagator as a sum of directed
random walks on a lattice is presented. Also a random walk representation for
the lattice fermionic propagators is developed for the case of the naive, the
Wilson, and the Kogut-Susskind fermions. For the naive fermions the phenomenon
of fermion doubling appears as having $2^{D}$ distinct spin factors being
associated with a single path in $D$-dimensions. In the case of the Wilson and
the Kogut-Susskind fermions, in the naive continuum limit, the path integral
representation coincides with the path integral representation for the
continuum fermionic propagator. Using this representation the Green's functions
of lattice QCD involving quark operators are written as a sum over the paths of
valence quark, the gauge fields and the sea quarks being integrated out.
Possible advantages of such a representation are illustrated by showing how one
can use numerical simulations to obtain a heuristic insight into the
relationship between QCD and the constituent quark model.
