Maximal inequalities and exponential estimates for stochastic convolutions driven by Levy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations

Zdzislaw Brzezniak; Jiahui Zhu; Wei Liu

doi:10.1137/18M1169011

Maximal inequalities and exponential estimates for stochastic convolutions driven by Levy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations

Zdzislaw Brzezniak, Jiahui Zhu, Wei Liu

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We present remarkably simple proofs of Burkholder–Davis–Gundy inequalities for
stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Levy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions
of Ito’s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Levy noise is established.

Original language	English
Pages (from-to)	2121–2167
Number of pages	47
Journal	SIAM journal on mathematical analysis
Early online date	23 May 2019
DOIs	https://doi.org/10.1137/18M1169011
Publication status	E-pub ahead of print - 23 May 2019

Bibliographical note

This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.

Keywords

Burkholder–Davis–Gundy inequality
maximal inequality
exponential estimate
stochastic convolution
Itˆo formula
martingale type r Banach space

Access to Document

10.1137/18M1169011

Maximal_inequalities_2019_03_07 (1)
This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.
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Cite this

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abstract = "We present remarkably simple proofs of Burkholder–Davis–Gundy inequalities forstochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Levy-type processes. Exponential estimates for stochastic convolutions are obtained and two versionsof Ito{\textquoteright}s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Levy noise is established.",

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AU - Zhu, Jiahui

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KW - maximal inequality

KW - exponential estimate

KW - stochastic convolution

KW - Itˆo formula

KW - martingale type r Banach space

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