  We give a probabilistic characterisation of the Besov-Lipschitz spaces
$Lip(\alpha,p,q)(X)$ on domains which support a Markovian kernel with
appropriate exponential bounds. This extends former results of
\cite{Jon,KPP1,KPP2,GHL} which were valid for $\alpha=\frac{d_w}{2},p=2$,
$q=\infty,$ where $d_w$ is the walk dimension of the space $X.$
