  The domain wall motion in a magnetic nanowire is examined theoretically in
the regime where the domain wall driving force is weak and its competition
against disorders is assisted by thermal agitations. Two types of driving
forces are considered; magnetic field and current. While the field induces the
domain wall motion through the Zeeman energy, the current induces the domain
wall motion by generating the spin transfer torque, of which effects in this
regime remain controversial. The spin transfer torque has two mutually
orthogonal vector components, the adiabatic spin transfer torque and the
nonadiabatic spin transfer torque. We investigate separate effects of the two
components on the domain wall depinning rate in one-dimensional systems and on
the domain wall creep velocity in two-dimensional systems, both below the
Walker breakdown threshold. In addition to the leading order contribution
coming from the field and/or the nonadiabatic spin transfer torque, we find
that the adiabatic spin transfer torque generates corrections, which can be of
relevance for an unambiguous analysis of experimental results. For instance, it
is demonstrated that the neglect of the corrections in experimental analysis
may lead to incorrect evaluation of the nonadiabaticity parameter. Effects of
the Rashba spin-orbit coupling on the domain wall motion are also analyzed.
