  We establish wavelet characterizations of homogeneous Besov spaces on
stratified Lie groups, both in terms of continuous and discrete wavelet
systems.
  We first introduce a notion of homogeneous Besov space $\dot{B}_{p,q}^s$ in
terms of a Littlewood-Paley-type decomposition, in analogy to the well-known
characterization of the Euclidean case. Such decompositions can be defined via
the spectral measure of a suitably chosen sub-Laplacian. We prove that the
scale of Besov spaces is independent of the precise choice of Littlewood-Paley
decomposition. In particular, different sub-Laplacians yield the same Besov
spaces.
  We then turn to wavelet characterizations, first via continuous wavelet
transforms (which can be viewed as continuous-scale Littlewood-Paley
decompositions), then via discretely indexed systems. We prove the existence of
wavelet frames and associated atomic decomposition formulas for all homogeneous
Besov spaces ${\dot B}_{p,q}^{s}$, with $1 \le p,q < \infty$ and $s \in
\mathbb{R}$.
