  Based on the recently proposed scenario of inflation driven by a bulk scalar
field in the braneworld of the Randall-Sundrum (RS) type, we investigate the
dynamics of a bulk scalar field on the inflating braneworld. We derive the late
time behavior of the bulk scalar field by analyzing the property of the
retarded Green function. We find that the late time behavior is basically
dominated by a single (or a pair of) pole(s) in the Green function irrespective
of the initial condition and of the signature of $m^{2}=V''(\phi)$, where
$V(\phi)$ is the potential of the bulk scalar field. Including the lowest order
back-reaction to the geometry, this late time behavior can be well approximated
by an effective 4-dimensional scalar field with $m^2_{\mathrm{eff}}=m^2/2$. The
mapping to the 4-dimensional effective theory is given by a simple scaling of
the potential with a redefinition of the field. Our result supports the picture
that the scenario of inflation driven by a bulk scalar field works in a quite
similar way to that in the standard 4-dimensional cosmology.
