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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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