arxiv: 2605.13727 · v1 · pith:KU374JYAnew · submitted 2026-05-13 · 🧮 math.PR

Stochastic evolution equations driven by arbitrary cylindrical L\'evy processes

Gergely Bod\'o , Sonja Cox , Adam Jakubowski , Markus Riedle This is my paper

Pith reviewed 2026-05-14 17:50 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic evolution equationscylindrical Lévy processesmild solutionsexistence and uniquenessEuler-Peano approximationHilbert spacespathwise convergence

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Mild solutions exist and are unique for abstract stochastic evolution equations driven by arbitrary cylindrical Lévy processes in Hilbert spaces.

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

The paper proves existence and uniqueness of mild solutions for stochastic evolution equations in Hilbert spaces when the driving noise is any cylindrical Lévy process. The coefficients satisfy only global Lipschitz conditions, and no moment or integrability assumptions are placed on the noise. The proof relies on a pathwise adaptive Euler-Peano scheme that uses stopping times adapted to individual noise paths together with a fixed-point argument for the mild solution map. A reader would care because this removes the need for semimartingale or Lévy-Itô decompositions that previously blocked treatment of generalized cylindrical noises in infinite dimensions.

Core claim

Under global Lipschitz conditions on the coefficients, the mild solution operator for abstract stochastic evolution equations in a Hilbert space driven by an arbitrary cylindrical Lévy process admits a unique fixed point. This fixed point is obtained as the limit of a pathwise Euler-Peano approximation constructed with noise-dependent stopping times, without invoking stochastic calculus that requires semimartingale structure.

What carries the argument

The pathwise adaptive Euler-Peano approximation scheme based on noise-dependent stopping times, which constructs candidate solutions directly from the cylindrical Lévy paths and feeds them into a fixed-point argument for the mild solution operator.

If this is right

The same approximation procedure yields the unique mild solution for any cylindrical Lévy noise without requiring finite moments.
Multiplicative noise is handled uniformly within the fixed-point framework.
The result supplies a constructive pathwise method that works in infinite-dimensional Hilbert spaces where classical Itô calculus does not apply.

Where Pith is reading between the lines

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

The stopping-time technique may extend to equations whose coefficients are only locally Lipschitz by localizing the approximations.
Numerical schemes derived from this construction could simulate solutions driven by infinite-variance Lévy noise without truncation.
The approach suggests a template for other generalized noises that lack semimartingale decompositions.

Load-bearing premise

The coefficients must satisfy global Lipschitz conditions so that the fixed-point map is a contraction and the approximations remain controlled.

What would settle it

An explicit globally Lipschitz pair of coefficients and a concrete cylindrical Lévy process for which the sequence of stopped Euler-Peano approximations fails to converge in the mild-solution topology would disprove the claim.

read the original abstract

We establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations driven by arbitrary cylindrical L\'evy processes in Hilbert spaces. The coefficients are assumed to satisfy global Lipschitz conditions, and no moment assumptions are imposed on the driving noise. The principal difficulty arises from the fact that cylindrical L\'evy processes exist solely in a generalised sense and typically admit no semimartingale or L\'evy-It\^o decomposition, which precludes the use of classical existence methods. To overcome these obstacles, we develop a pathwise adaptive Euler-Peano approximation scheme based on noise-dependent stopping times and a fixed-point formulation of the mild solution operator. The resulting approach avoids stochastic calculus techniques relying on semimartingale decompositions and provides a robust and flexible framework for treating multiplicative cylindrical L\'evy noise in infinite-dimensional systems.

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.

Desk Editor's Note private letter to a colleague

First general existence result for mild solutions to SEEs driven by arbitrary cylindrical Lévy processes via a new noise-adaptive Euler-Peano scheme, but the no-moment convergence step needs verification.

read the letter

This paper establishes the first existence and uniqueness theorem for mild solutions to abstract stochastic evolution equations driven by arbitrary cylindrical Lévy processes in Hilbert spaces, with no moment assumptions on the noise. The key is a new pathwise adaptive Euler-Peano approximation using noise-dependent stopping times to truncate jumps, followed by a fixed-point argument for the mild solution operator. What stands out is how they sidestep the lack of semimartingale or Lévy-Itô decompositions that block classical methods. The scheme adapts to the specific realization of the cylindrical process, which is a practical way to handle the generalized nature of these noises. Global Lipschitz conditions on the coefficients make the fixed-point contraction work. The method looks promising for extending SPDE theory to more general noises. It could apply to models where standard Lévy assumptions fail. The soft spot is in the convergence of the approximations. Without moments, it's not obvious that the error between stopped versions stays controlled uniformly as the truncation level goes to infinity. The stress-test note flags that Burkholder-type bounds might not hold, so the limit passage needs tight estimates. Since the abstract claims it works, the full paper must have the details, but that's where I'd focus scrutiny. This work is aimed at specialists in infinite-dimensional stochastic analysis. It fills a notable gap, so it deserves peer review to check the technical steps. I'd send it out for refereeing.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations in Hilbert spaces driven by arbitrary cylindrical Lévy processes. The coefficients satisfy global Lipschitz conditions with no moment assumptions imposed on the noise. The approach relies on a pathwise adaptive Euler-Peano approximation scheme using noise-dependent stopping times to truncate large jumps, combined with a fixed-point formulation of the mild solution operator, thereby avoiding classical stochastic calculus that requires semimartingale decompositions.

Significance. If the convergence arguments hold, the result would constitute a substantial extension of the theory of infinite-dimensional stochastic equations, as it accommodates fully general cylindrical Lévy noises (including those with infinite variation or heavy tails) that lack standard Lévy-Itô decompositions or moment bounds. This could enable new applications in systems where previous results were restricted by integrability requirements on the driving process.

major comments (2)

[Proof of Theorem 3.1 (main existence result)] The core convergence claim for the adaptive Euler-Peano scheme (detailed in the proof of the main existence theorem) requires uniform control of the stopped increments in the Hilbert-space norm without any moment assumptions on the cylindrical Lévy process. It is unclear whether the error estimates between successive approximations close uniformly in the truncation level, since standard Burkholder-type bounds or integrability for the stochastic convolution may fail when the Lévy measure has infinite variation; this directly affects the passage to the limit in the mild-solution fixed-point space.
[Section 4, fixed-point formulation] In the fixed-point argument for the mild solution operator (Section 4), the contraction constant is asserted to remain strictly less than 1 independently of the sequence of stopping times. However, the estimates do not explicitly verify that the Lipschitz constants of the coefficients and the semigroup bounds yield a uniform contraction factor across all truncations, which is load-bearing for the completeness of the Cauchy sequence in the solution space.

minor comments (2)

[Introduction] The introduction would benefit from a brief explicit comparison table or paragraph contrasting the new scheme with prior results that impose finite-moment conditions on the Lévy noise.
[Preliminaries] Notation for the cylindrical Lévy process (e.g., the precise meaning of 'arbitrary' and the generalized sense in which it exists) should be introduced with a dedicated preliminary subsection before the approximation scheme is defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major points below and will revise the manuscript to provide additional clarifications and explicit verifications in the proof of Theorem 3.1 and Section 4.

read point-by-point responses

Referee: [Proof of Theorem 3.1 (main existence result)] The core convergence claim for the adaptive Euler-Peano scheme (detailed in the proof of the main existence theorem) requires uniform control of the stopped increments in the Hilbert-space norm without any moment assumptions on the cylindrical Lévy process. It is unclear whether the error estimates between successive approximations close uniformly in the truncation level, since standard Burkholder-type bounds or integrability for the stochastic convolution may fail when the Lévy measure has infinite variation; this directly affects the passage to the limit in the mild-solution fixed-point space.

Authors: We appreciate the referee pointing out the need for greater clarity on uniformity. Our adaptive stopping times are constructed pathwise from the jumps of the given cylindrical Lévy process, so that on each stopped interval the driving noise has jumps of size at most 1/n. This permits direct, pathwise Gronwall-type estimates that use only the global Lipschitz constant of the coefficients and the uniform bound on the semigroup; no Burkholder–Davis–Gundy inequality or moment assumption on the untruncated process is invoked. The error between successive approximations is shown to be controlled by a quantity that tends to zero as n→∞ uniformly in the truncation level because the stopping times increase to infinity and the fixed-point space is equipped with a norm that absorbs the stopped increments. We will insert a new lemma and expanded estimates in the revised proof of Theorem 3.1 to make this uniformity explicit. revision: yes
Referee: [Section 4, fixed-point formulation] In the fixed-point argument for the mild solution operator (Section 4), the contraction constant is asserted to remain strictly less than 1 independently of the sequence of stopping times. However, the estimates do not explicitly verify that the Lipschitz constants of the coefficients and the semigroup bounds yield a uniform contraction factor across all truncations, which is load-bearing for the completeness of the Cauchy sequence in the solution space.

Authors: We agree that an explicit check of independence would improve readability. The mild-solution operator is defined for each truncated noise using the same Lipschitz constant L of the coefficients and the same semigroup bound M. For any fixed time horizon T small enough that L M T < 1, the contraction factor is therefore strictly less than 1 and does not depend on the particular sequence of stopping times. The same factor applies uniformly to all truncations because the operator is constructed identically once the driving noise is replaced by its stopped version. We will add a short paragraph immediately after the statement of the contraction mapping in Section 4 that records this independence explicitly. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result depends on standard Hilbert-space mild-solution theory and the global Lipschitz assumption to close the fixed-point argument; no free parameters or new entities are introduced.

axioms (2)

domain assumption Global Lipschitz continuity of the coefficients
Invoked to guarantee the contraction mapping property for the mild solution operator.
standard math Existence of mild solutions in the Hilbert-space setting
Background assumption from the theory of stochastic evolution equations.

pith-pipeline@v0.9.0 · 5445 in / 1244 out tokens · 61632 ms · 2026-05-14T17:50:55.999657+00:00 · methodology

discussion (0)

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