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

Abstract : We show that any R^d \{0}-valued self-similar Markov process X, with index α > 0 can be represented as a path transformation of some Markov additive process (MAP) (θ, ξ) in S_{d−1} × R. This result extends the well known Lamperti transformation. Let us denote by X the self-similar Markov process which is obtained from the MAP (θ, −ξ) through this extended Lamperti transformation. Then we prove that X is in weak duality with X, with respect to the measure π(x/|x|)|x|^{α−d}dx, if and only if (θ, ξ) is reversible with respect to the measure π(ds)dx, where π(ds) is some σ-finite measure on S_{d−1} and dx is the Lebesgue measure on R. Besides, the dual process X has the same law as the inversion (X_{γ_t}/|X|_{γ_t}^2, t ≥ 0) of X, where γ t is the inverse of t → \int_0^t 0|X|^{−2α}_s ds. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.
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https://hal.archives-ouvertes.fr/hal-01266723
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Submitted on : Wednesday, February 3, 2016 - 11:39:59 AM
Last modification on : Monday, March 9, 2020 - 6:15:53 PM
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Larbi Alili, Loïc Chaumont, Piotr Graczyk, Tomasz Żak. Inversion, duality and Doob h-transforms for self-similar Markov processes. 2016. ⟨hal-01266723⟩

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