  We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov
process $X$, with index $\alpha>0$ can be represented as a path transformation
of some Markov additive process (MAP) $(\theta,\xi)$ in
$S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti
transformation. Let us denote by $\widehat{X}$ the self-similar Markov process
which is obtained from the MAP $(\theta,-\xi)$ through this extended Lamperti
transformation. Then we prove that $\widehat{X}$ is in weak duality with $X$,
with respect to the measure $\pi(x/\|x\|)\|x\|^{\alpha-d}dx$, if and only if
$(\theta,\xi)$ is reversible with respect to the measure $\pi(ds)dx$, where
$\pi(ds)$ is some $\sigma$-finite measure on $S_{d-1}$ and $dx$ is the Lebesgue
measure on $\mathbb{R}$. Besides, the dual process $\widehat{X}$ has the same
law as the inversion $(X_{\gamma_t}/\|X_{\gamma_t}\|^2,t\ge0)$ of $X$, where
$\gamma_t$ is the inverse of $t\mapsto\int_0^t\|X\|_s^{-2\alpha}\,ds$. These
results allow us to obtain excessive functions for some classes of self-similar
Markov processes such as stable L\'evy processes.
