  We show existence and uniqueness of a stationary state for a kinetic
Fokker-Planck equation modelling the fibre lay-down process in the production
of non-woven textiles. Following a micro-macro decomposition, we use
hypocoercivity techniques to show exponential convergence to equilibrium with
an explicit rate assuming the conveyor belt moves slow enough. This work is an
extension of (Dolbeault et al., 2013), where the authors consider the case of a
stationary conveyor belt. Adding the movement of the belt, the global Gibbs
state is not known explicitly. We thus derive a more general hypocoercivity
estimate from which existence, uniqueness and exponential convergence can be
derived. To treat the same class of potentials as in (Dolbeault et al., 2013),
we make use of an additional weight function following the Lyapunov functional
approach in (Kolb et al., 2013).
