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arxiv: 1907.11867 · v1 · pith:WNQM5MMMnew · submitted 2019-07-27 · 🧮 math.PR · math.AP

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

Jiahui Zhu , Zdzis{\l}aw Brze\'zniak , Wei Liu This is my paper

Pith reviewed 2026-05-24 14:55 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords Lévy-type processesstochastic convolutionsmaximal inequalitiesBanach spacesquasi-geostrophic equationsmild solutionsBurkholder-Davis-Gundy inequalities
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The pith

Maximal inequalities for Lévy-driven stochastic convolutions in Banach spaces prove unique mild solutions for the stochastic quasi-geostrophic equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors give simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions driven by Lévy-type processes in Banach spaces. They also obtain exponential estimates for the convolutions and derive two versions of Itô's formula. These maximal inequalities are then used to establish the existence and uniqueness of mild solutions to the stochastic quasi-geostrophic equation driven by Lévy noise. This matters because it provides tools for analyzing stochastic fluid equations with more general noise types in infinite-dimensional spaces.

Core claim

The paper establishes Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes through simple proofs. Exponential estimates are derived for these convolutions, along with two versions of Itô's formula in Banach spaces. The maximal inequality is applied to demonstrate the existence and uniqueness of mild solutions for the stochastic quasi-geostrophic equation with Lévy noise.

What carries the argument

The maximal inequality for stochastic convolutions driven by Lévy-type processes, which provides bounds on the supremum norm of the convolution process.

If this is right

  • Existence and uniqueness of mild solutions to the stochastic quasi-geostrophic equation with Lévy noise.
  • Exponential estimates hold for the stochastic convolutions in the Banach space setting.
  • Two versions of Itô's formula are valid in Banach spaces for these processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The inequalities could be applied to establish well-posedness for other stochastic partial differential equations with Lévy noise.
  • These estimates might facilitate the study of long-time behavior or invariant measures for the quasi-geostrophic equation.
  • Extensions to time-dependent or nonlinear coefficients in the driving processes could be explored based on the proof techniques.

Load-bearing premise

The specific conditions on the Lévy-type process and the Banach space must hold so that the maximal inequality applies directly to the mild solution of the quasi-geostrophic equation.

What would settle it

Observing a case where a stochastic convolution driven by a Lévy-type process in a Banach space exceeds the bound given by the maximal inequality, or where multiple mild solutions exist for the stochastic quasi-geostrophic equation.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript derives Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes, along with exponential estimates and two versions of Itô's formula. It then applies the maximal inequality to prove existence and uniqueness of mild solutions to the stochastic quasi-geostrophic equation with Lévy noise.

Significance. If the derivations hold under the stated assumptions on the Lévy-type process and Banach space, the work supplies concrete analytic tools for SPDEs with jump noise. The explicit application to existence/uniqueness for the quasi-geostrophic equation is a direct, falsifiable contribution that links the inequalities to a physically motivated model.

minor comments (2)
  1. The abstract states that the inequalities are derived under 'stated assumptions'; the introduction or §2 should include a compact table or list summarizing the precise conditions on the Lévy measure, the Banach space, and the semigroup that are used throughout.
  2. Notation for the stochastic convolution and the mild solution in the final section could be aligned more explicitly with the notation introduced for the inequalities (e.g., consistent use of the same symbols for the integrability exponents).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes, then obtains exponential estimates and applies the maximal inequality to prove existence and uniqueness of mild solutions for the stochastic quasi-geostrophic equation. All steps rely on standard stochastic analysis techniques under explicitly stated assumptions on the Lévy-type process and Banach space; no derivation reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain. The application section uses the derived inequalities directly on the mild formulation without circular reduction. This is the normal case of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard axioms of stochastic integration in Banach spaces and properties of Lévy processes; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of Lévy processes and stochastic integrals in Banach spaces hold under the conditions needed for BDG inequalities.
    Invoked to derive the maximal inequalities and apply them to the SPDE.

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