A modified Φ-Sobolev inequality for canonical Lévy processes and its applications
Volume 10, Issue 2 (2023), pp. 145–173
Pub. online: 23 January 2023
Type: Research Article
Open Access
Open Access
Received
10 June 2022
10 June 2022
Revised
15 November 2022
15 November 2022
Accepted
11 January 2023
11 January 2023
Published
23 January 2023
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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