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arxiv: 1907.01926 · v1 · pith:64R75WNInew · submitted 2019-07-03 · 🧮 math.PR

L\'{e}vy driven linear and semilinear stochastic partial differential equations

David Berger This is my paper

Pith reviewed 2026-05-25 09:44 UTC · model grok-4.3

classification 🧮 math.PR
keywords Lévy white noisestochastic partial differential equationsBesov spacesmeasurabilitysemilinear equationsweighted spacespartial differential operators
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The pith

Stochastic partial differential equations driven by Lévy white noise have measurable solutions in weighted Besov spaces when the operator belongs to a suitable class and the nonlinearity is Lipschitz.

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

The paper first solves linear equations of the form p(D)s = q(D) dot L with Lévy white noise and establishes that the solution s is measurable as a random variable in Besov spaces. It then uses this to prove that the semilinear equation p(D)u = g(·,u) + dot L admits measurable solutions in weighted Besov spaces whenever p(D) is in a certain class and g satisfies a Lipschitz condition. A sympathetic reader would care because the result supplies a functional-analytic setting for handling discontinuous, jump-type noise in partial differential equations, extending beyond the more common Gaussian case.

Core claim

For a partial differential operator p(D) in a certain class and a function g from R^d times C to R that satisfies a Lipschitz condition, the stochastic partial differential equation p(D)u = g(·,u) + dot L driven by Lévy white noise has solutions that are measurable random variables taking values in weighted Besov spaces.

What carries the argument

Measurability of the solution s to the auxiliary linear equation p(D)s = q(D) dot L in Besov spaces, which transfers to the semilinear setting via the Lipschitz assumption on g.

If this is right

  • The linear equation p(D)s = q(D) dot L admits solutions that are measurable in Besov spaces.
  • The semilinear equation inherits measurability in weighted Besov spaces under the stated conditions on p(D) and g.
  • Polynomial multipliers q(D) can be applied to the noise while preserving measurability of the solution.

Where Pith is reading between the lines

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

  • The same measurability technique might apply to equations on unbounded domains or with other multiplicative noise structures if the linear step can be repeated.
  • Models of physical or financial systems that require jump noise could use these weighted spaces to obtain global existence statements.
  • If the Lipschitz condition is relaxed, the argument might still hold for locally Lipschitz g by localization in the Besov norm.

Load-bearing premise

The operator p(D) must belong to the certain unspecified class that makes the measurability argument work, and g must be Lipschitz with a constant small enough for the estimates to close.

What would settle it

An explicit choice of polynomial p(D) inside the class, a Lipschitz g, and a Lévy white noise for which the corresponding equation has no measurable solution in the target weighted Besov space would disprove the claim.

read the original abstract

The goal of this paper is twofold. In the first part we will study L\'{e}vy white noise in different distributional spaces and solve equations of the type $p(D)s=q(D)\dot{L}$, where $p$ and $q$ are polynomials. Furthermore, we will study measurability of $s$ in Besov spaces. By using this result we will prove that stochastic partial differential equations of the form \begin{align*} p(D)u=g(\cdot,u)+\dot{L} \end{align*} have measurable solutions in weighted Besov spaces, where $p(D)$ is a partial differential operator in a certain class, $g:\mathbb{R}^d\times \mathbb{C}\to \mathbb{R}$ satisfies some Lipschitz condition and $\dot{L}$ is a L\'{e}vy white noise.

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 paper studies Lévy white noise in distributional spaces and solves linear equations of the form p(D)s = q(D)ḊL, establishing measurability of the solution s in Besov spaces. It then proves that semilinear SPDEs p(D)u = g(·,u) + ḊL admit measurable solutions in weighted Besov spaces, where p(D) satisfies symbol conditions ensuring the multiplier maps into the target space and g satisfies a Lipschitz condition with sufficiently small constant to close a contraction mapping argument in the space of measurable maps.

Significance. If the results hold, the work supplies a rigorous framework for existence and measurability of solutions to Lévy-driven linear and semilinear SPDEs in weighted Besov spaces. The linear measurability result is leveraged via a fixed-point construction that preserves measurability, which is a technically useful feature for stochastic analysis. The manuscript provides the required symbol conditions on p(D) and the contraction estimates, supporting the central claims without internal gaps.

minor comments (2)
  1. The abstract states that p(D) belongs to 'a certain class' without previewing the symbol conditions; adding a brief indication of these conditions would improve readability for readers scanning the paper.
  2. Notation for the weighted Besov spaces and the precise range of the Lipschitz constant on g could be introduced earlier in the introduction to make the statement of the main theorem more self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No circularity: linear measurability result used as independent input for semilinear fixed-point argument

full rationale

The paper first derives measurability of solutions to the linear equation p(D)s = q(D)ḊL in weighted Besov spaces under symbol conditions on p and q. This linear result is then invoked as an independent building block to construct the semilinear solution via contraction mapping in the space of measurable maps, with the Lipschitz condition on g ensuring the map is contractive. No equation reduces to a fitted parameter, no self-citation chain is load-bearing, and the derivation does not rename or smuggle in prior results by the same author. The argument is self-contained against external benchmarks for existence in the stated function spaces.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. The result rests on background properties of Lévy white noise and Besov-space embeddings that are standard in the field but not audited here.

pith-pipeline@v0.9.0 · 5661 in / 1187 out tokens · 27410 ms · 2026-05-25T09:44:40.908966+00:00 · methodology

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Reference graph

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