  The dissipative dynamics of a vortex line in a superfluid is investigated
within the frame of a non-Markovian quantal Brownian motion model. Our starting
point is a recently proposed interaction Hamiltonian between the vortex and the
superfluid quasiparticle excitations, which is generalized to incorporate the
effect of scattering from fermion impurities ($^3$He atoms). Thus, a
non-Markovian equation of motion for the mean value of the vortex position
operator is derived within a weak-coupling approximation. Such an equation is
shown to yield, in the Markovian and elastic scattering limits, a $^3$He
contribution to the longitudinal friction coefficient equivalent to that
arising from the Rayfield-Reif formula. Simultaneous Markov and elastic
scattering limits are found, however, to be incompatible, since an unexpected
breakdown of the Markovian approximation is detected at low cyclotron
frequencies. Then, a non-Markovian expression for the longitudinal friction
coefficient is derived and computed as a function of temperature and $^3$He
concentration. Such calculations show that cyclotron frequencies within the
range 0.01$-$0.03 ps$^{-1}$ yield a very good agreement to the longitudinal
friction figures computed from the Iordanskii and Rayfield-Reif formulas for
pure $^4$He, up to temperatures near 1 K. A similar performance is found for
nonvanishing $^3$He concentrations, where the comparison is also shown to be
very favorable with respect to the available experimental data. Memory effects
are shown to be weak and increasing with temperature and concentration.
