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

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Alili, Larbi, Chaumont, Loïc, Graczyk, Piotr and Żak, Tomasz (2017) Inversion, duality and Doob h-transforms for self-similar Markov processes. Electronic Journal of Probability, 22 . 20. doi:10.1214/17-EJP33 ISSN 1083-6489.

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Official URL: http://dx.doi.org/10.1214/17-EJP33

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Abstract

We show that any Rd∖{0}Rd∖{0}-valued self-similar Markov process XX, with index α>0α>0 can be represented as a path transformation of some Markov additive process (MAP) (θ,ξ)(θ,ξ) in Sd−1×RSd−1×R. This result extends the well known Lamperti transformation. Let us denote by XˆX^ the self-similar Markov process which is obtained from the MAP (θ,−ξ)(θ,−ξ) through this extended Lamperti transformation. Then we prove that XˆX^ is in weak duality with XX, with respect to the measure π(x/∥x∥)∥x∥α−ddxπ(x/‖x‖)‖x‖α−ddx, if and only if (θ,ξ)(θ,ξ) is reversible with respect to the measure π(ds)dxπ(ds)dx, where π(ds)π(ds) is some σσ-finite measure on Sd−1Sd−1 and dxdx is the Lebesgue measure on RR. Moreover, the dual process XˆX^ has the same law as the inversion (Xγt/∥Xγt∥2,t≥0)(Xγt/‖Xγt‖2,t≥0) of XX, where γtγt is the inverse of t↦∫t0∥X∥−2αsdst↦∫0t‖X‖s−2αds. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH): Markov processes, Brownian motion processes
Journal or Publication Title: Electronic Journal of Probability
Publisher: University of Washington. Dept. of Mathematics
ISSN: 1083-6489
Official Date: 18 February 2017
Dates:
DateEvent
18 February 2017Published
30 January 2017Accepted
22 March 2016Submitted
Volume: 22
Number of Pages: 18
Article Number: 20
DOI: 10.1214/17-EJP33
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 9 March 2017
Date of first compliant Open Access: 8 May 2019
Funder: Narodowe Centrum Nauki [Polish National Science Center]
Grant number: Project no. 2015/17/B/ST1/01043
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