# Maximal inequalities and exponential estimates for stochastic convolutions driven by L\'{e}vy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations

## TL;DR

This paper develops new inequalities and exponential estimates for stochastic convolutions driven by Lévy processes in Banach spaces, and applies these results to establish solutions for stochastic quasi-geostrophic equations.

## Contribution

It provides simplified proofs of key inequalities and derives exponential estimates, enabling the analysis of stochastic quasi-geostrophic equations with Lévy noise.

## Key findings

- Maximal inequalities for stochastic convolutions established
- Exponential estimates derived for stochastic convolutions
- Existence and uniqueness of solutions for stochastic quasi-geostrophic equations proven

## 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 L\'{e}vy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of It\^{o}'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 L\'{e}vy noise is established.