  We consider a 3D homogeneous superfluid at low temperature $T$ with 2 types
of excitations, gapless phonons with a linear dispersion relation at low
wavenumber, and gapped quasiparticles with a quadratic dispersion relation
around extrema. We calculate the scattering amplitude of a phonon on a
quasiparticle to leading order in $T$ for all subsonic quasiparticle
velocities, with a $S$-matrix formalism between exact asymptotic states dressed
by virtual phonons. We then characterize the erratic motion of the
quasiparticle in the superfluid due to its unceasing collisions with thermal
phonons through mean force $F(k)$, longitudinal and transverse $k$-dependent
momentum diffusion coefficients, and spatial diffusion coefficient. At the
minimum location $k_0$ of the dispersion relation, where the velocity vanishes,
$F(k)$ varies linearly with velocity with an isotropic friction coefficient; if
$k_0=0$, the momentum diffusion is also isotropic and $F(k_0)=0$; if $k_0>0$,
it is not, and $F(k_0)$ is nonzero but subleading with respect to friction by
one order in $T$. The velocity time correlation function, whose integral is the
spatial diffusion coefficient, decays with the mean velocity damping rate if
$k_0=0$; if $k_0>0$, it has a second exponential component, with an amplitude
and a damping rate lower by a factor $\propto T$ (it is the velocity direction
thermalization rate). We also characterize force and momentum diffusion close
to the stability domain sonic edge. Our general expressions are expected to be
exact to leading order in $T$. We illustrate them in the BCS approximation, for
a fermionic quasiparticle (an unpaired fermion) in a superfluid of spin 1/2
fermions, realisable with cold atoms in flat bottom traps. We also refute the
statement of Lerch, Bartosch and Kopietz (2008), that there would be no
fermionic quasiparticle in such a superfluid.
