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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
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Official URL: http://dx.doi.org/10.1214/17-EJP33
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 | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of 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: |
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| Date of first compliant deposit: | 9 March 2017 | ||||||||
| 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 | ||||||||
| Funder: | Narodowe Centrum Nauki [Polish National Science Center] | ||||||||
| Grant number: | Project no. 2015/17/B/ST1/01043 | ||||||||
| Adapted As: |
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