## Inversion, duality and Doob $$h$$-transforms for self-similar Markov processes.(English)Zbl 1357.60079

Summary: 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 $$\mathbb{R}$$. Moreover, the dual process $$\widehat{X}$$ has the same law as the inversion $$(X_{\gamma_t}/\|X_{\gamma_t}\|^2,t\geq 0)$$ 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 Lévy processes.

### MSC:

 60J25 Continuous-time Markov processes on general state spaces 60J45 Probabilistic potential theory 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 60J65 Brownian motion 60J75 Jump processes (MSC2010)
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