Inversion, duality and Doob $h$-transforms for self-similar Markov processes

Abstract

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 $\hat{X}$ the self-similar Markov process which is obtained from the MAP $(\theta ,-\xi )$ through this extended Lamperti transformation. Then we prove that $\hat{X}$ is in weak duality with $X$, with respect to the measure $\pi (x/\Vert x\Vert )\Vert x{\Vert }^{\alpha -d}dx$, 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 $\hat{X}$ has the same law as the inversion $(X_{{\gamma }_t}/\Vert X_{{\gamma }_t}{\Vert }^2,t\ge 0)$ of $X$, where ${\gamma }_t$ is the inverse of $t\mapsto {\int }_0^t\Vert X{\Vert }_s^{-2\alpha }ds$. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.