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

Larbi Alili (1) , Loïc Chaumont (2) , Piotr Graczyk (2) , Tomasz Żak (3)

1 Department of Mathematics, University of Warwick
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
3 Technical university of Wroclaw

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.

Mots clés: Self-similar Markov processes; Markov additive processes; time change; inversion; duality; Doob h-transform.

Domaines: Probabilités [math.PR]

Larbi Alili, Loïc Chaumont, Piotr Graczyk, Tomasz Żak. Inversion, duality and Doob h-transforms for self-similar Markov processes. 2016. ⟨hal-01266723⟩