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A modified Φ-Sobolev inequality for canonical Lévy processes and its applications
Volume 10, Issue 2 (2023), pp. 145–173
Noriyoshi Sakuma ORCID icon link to view author Noriyoshi Sakuma details   Ryoichi Suzuki ORCID icon link to view author Ryoichi Suzuki details  

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https://doi.org/10.15559/23-VMSTA220
Pub. online: 23 January 2023      Type: Research Article      Open accessOpen Access

Received
10 June 2022
Revised
15 November 2022
Accepted
11 January 2023
Published
23 January 2023

Abstract

A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.

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Keywords
Malliavin calculus Lévy processes logarithmic Sobolev inequalities Φ-Sobolev inequalities deviation inequalities concentration inequalities

MSC2010
60H07 60G51 60J75 60E15

Funding
This work was supported by JSPS Open Partnership Joint Research Projects grant JPJSBP120209921, JPJSBP120203202, KAKENHI JP19K03515 and JP19H01791.

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