Probabilistic characterisation of Besov-Lipschitz spaces on metric measure spaces

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 $\alpha =\frac{d_w}{2},p=2$, $q=\mathrm{\infty },$ where $d_w$ is the walk dimension of the space $X\mathrm{.}$