Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

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 Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô'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 Lévy noise is established.

Lp-solutions for stochastic Navier–Stokes equations with jump noise

“…replace L 4 (D) by H −2α,4 (D). Since L p (D) spaces for p ≥ 2 are martingale type 2 Banach spaces, one can define the stochastic integral of P ⊗ Z-measurable function ξ in the martingale type 2 Banach space setting, see [3,25]. Recall that the stochastic integral an H −2α,4 (D)-valued càdlàg modification and thus we do not require that from the solution from Definition 2.6 to be a càdlàg H −2α,4 (D)-valued process.…”

“…Since L 4 (R + ; L 4 sol (D)) is martingale type 2 Banach space and L 4 (0, T ; L 4 sol (D)) can be isometrically identified with a closed subspace of L 4 (R + ; L 4 sol (D)), therefore L 4 (0, T ; L 4 sol (D)) is also martingale type 2. Now applying the Burkholder's inequality in Banach space (see [3,25]), we obtain Inserting back gives that…”

“…Remark 3.1. For the quasi-geostrophic equation, (3.5) follows from maximal inequality established in [25].…”

“…In comparison to [4], where solutions are in L ∞ (0, T ; L 2 (D))∩L 2 (0, T ; H 1 0 (D)) and the OU process Z is assumed to be in L 2 (0, T ; H 1 0 (D)), we work for equations with much weaker assumptions on the noise than the Galerkin approximation method could handle. The fundamental ideas of the proofs are inspired by our recent work [25]. We established in [25] a type of maximal inequality for stochastic convolutions driven by Lévy-type noise and as an application, we showed the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equations in terms of L p theory.…”

“…The fundamental ideas of the proofs are inspired by our recent work [25]. We established in [25] a type of maximal inequality for stochastic convolutions driven by Lévy-type noise and as an application, we showed the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equations in terms of L p theory. However, since (e −tAp ) is only a bounded analytic C 0 -semigroup in L p (D), we cannot apply the maximal inequality for C 0 -contraction semigroups in [25].…”