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

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.