Electronic Journal of Probability

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

Larbi Alili, Loïc Chaumont, Piotr Graczyk, and Tomasz Żak

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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 $\newcommand{\ed } {\stackrel{(d)} {=}} \pi (ds)dx$, 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 $(X_{\gamma _t}/\|X_{\gamma _t}\|^2,t\ge 0)$ 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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 20, 18 pp.

Received: 22 March 2016
Accepted: 30 January 2017
First available in Project Euclid: 18 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 31C05: Harmonic, subharmonic, superharmonic functions
Secondary: 60J65: Brownian motion [See also 58J65] 60J75: Jump processes

self-similar Markov processes Markov additive processes time change inversion duality Doob $h$-transform

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Alili, Larbi; Chaumont, Loïc; Graczyk, Piotr; Żak, Tomasz. Inversion, duality and Doob $h$-transforms for self-similar Markov processes. Electron. J. Probab. 22 (2017), paper no. 20, 18 pp. doi:10.1214/17-EJP33. https://projecteuclid.org/euclid.ejp/1487386998

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