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arxiv: 2604.12317 · v1 · submitted 2026-04-14 · 🧮 math.PR

Weak solution for distribution dependent SDEs driven by L\'{e}vy noise

Mingkun Ye This is my paper

Pith reviewed 2026-05-10 15:16 UTC · model grok-4.3

classification 🧮 math.PR
keywords weak solutionsdistribution-dependent SDEsLévy noiseKrylov estimatetightness argumentstochastic differential equationsexistence
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The pith

Weak solutions exist for distribution-dependent SDEs driven by Lévy noise when the drifts satisfy specific integrability conditions.

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

The paper proves that distribution-dependent stochastic differential equations driven by a broad class of Lévy noises admit weak solutions as long as the drift coefficients obey certain integrability conditions. The argument first derives a Krylov-type estimate to obtain the required controls on the distribution-dependent terms and then applies a tightness argument to extract a convergent subsequence from suitable approximations. If this holds, models of interacting systems with jumps—such as particle populations or asset prices subject to sudden moves—gain a rigorous existence foundation without needing stronger regularity on the coefficients. A reader interested in stochastic modeling would care because the result broadens the class of noises for which such mean-field-type equations are well-posed.

Core claim

We establish the existence of weak solutions for distribution-dependent stochastic differential equations (DDSDEs) driven by a broad class of Lévy noises, where the drift coefficients satisfy specific integrability conditions. This is achieved through the Krylov-type estimate and tightness argument.

What carries the argument

Krylov-type estimate that produces the necessary integrability bounds for the distribution-dependent drift, used together with tightness of the laws of approximating solutions in the space of càdlàg paths.

Load-bearing premise

The drift coefficients must satisfy the specific integrability conditions that make the Krylov-type estimate applicable.

What would settle it

A concrete example of a distribution-dependent SDE driven by Lévy noise whose drift meets the stated integrability conditions but whose approximating sequence fails to be tight or converges to a process that does not solve the equation.

read the original abstract

In this paper, we establish the existence of weak solutions for distribution-dependent stochastic differential equations (DDSDEs) driven by a broad class of L\'{e}vy noises, where the drift coefficients satisfy specific integrability conditions. This is achieved through the Krylov-type estimate and tightness argument.

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

1 major / 1 minor

Summary. The manuscript establishes the existence of weak solutions for distribution-dependent SDEs (DDSDEs) driven by a broad class of Lévy noises. The drift coefficients are assumed to satisfy specific integrability conditions; the proof proceeds by deriving Krylov-type estimates to control the distribution-dependent drift integral, establishing tightness of the approximating laws, and extracting a weakly convergent subsequence whose limit is asserted to solve the DDSDE.

Significance. If the central argument holds, the result would extend known weak-existence theorems for DDSDEs from Brownian drivers to Lévy processes under comparatively weak integrability assumptions on the coefficients. The reliance on Krylov estimates and tightness is a standard and potentially robust approach that could apply to many Lévy measures.

major comments (1)
  1. [Tightness and limit identification argument] The passage-to-the-limit step for the distribution-dependent drift term (typically in the section deriving the limit equation from the tightness argument) rests on an unverified continuity property. Pure integrability of b(t,x,μ) (e.g., L^p or L^1 bounds) does not automatically yield that ∫ b(t,X_t,μ_n) dt → ∫ b(t,X_t,μ) dt when μ_n converges weakly to μ; an additional structural hypothesis (Lipschitz continuity in Wasserstein distance, continuity in the weak topology, or uniform integrability with respect to the limiting measure) is required for the identification of the limit as a solution of the DDSDE. The manuscript must either add and verify such a condition or demonstrate that the stated integrability alone suffices.
minor comments (1)
  1. The abstract is terse and does not state the precise integrability conditions on the drift; including them would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the point regarding the passage to the limit for the distribution-dependent drift term below.

read point-by-point responses

  1. Referee: The passage-to-the-limit step for the distribution-dependent drift term (typically in the section deriving the limit equation from the tightness argument) rests on an unverified continuity property. Pure integrability of b(t,x,μ) (e.g., L^p or L^1 bounds) does not automatically yield that ∫ b(t,X_t,μ_n) dt → ∫ b(t,X_t,μ) dt when μ_n converges weakly to μ; an additional structural hypothesis (Lipschitz continuity in Wasserstein distance, continuity in the weak topology, or uniform integrability with respect to the limiting measure) is required for the identification of the limit as a solution of the DDSDE. The manuscript must either add and verify such a condition or demonstrate that the stated integrability alone suffices.

    Authors: We agree that the identification of the limit for the integral term ∫ b(t,X_t,μ_n) dt requires justification beyond mere weak convergence of the laws. The Krylov-type estimates derived in the paper are designed specifically to control the distribution-dependent drift and, in fact, yield uniform integrability of the family of integrands with respect to the approximating measures. This uniform integrability, together with the tightness and the weak convergence of the processes, permits passage to the limit via the Vitali convergence theorem (or an analogous result in the appropriate L^1 space). We will revise the manuscript to make this step fully explicit, including a dedicated paragraph that invokes the uniform integrability coming from the Krylov bound and confirms that no additional continuity assumption on b in the measure variable is needed under our stated integrability hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity: direct existence proof via tightness and Krylov estimates

full rationale

The paper establishes weak solution existence for distribution-dependent SDEs with Lévy noise using standard Krylov estimates and tightness. No equations, parameters, or self-citations appear in the abstract or description that would make the result reduce to its inputs by construction. The approach relies on classical stochastic analysis methods (Krylov bounds for integrability, Prohorov tightness for weak convergence), which are independent of the paper's own fitted values or prior self-referential claims. This is a self-contained proof, not a tautology or renamed fit. The skeptic concern about measure continuity is a potential correctness gap but does not indicate circularity per the defined patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background results for Lévy processes and SDEs; the key unverified step is the applicability of Krylov-type estimates under the stated integrability.

axioms (2)
  • standard math Lévy processes exist with the given Lévy measure and satisfy standard integrability for the stochastic integral
    Invoked implicitly when defining the DDSDE driven by Lévy noise.
  • domain assumption Krylov-type estimates hold for the approximating processes under the integrability conditions on the drift
    Central to the proof strategy described in the abstract.

pith-pipeline@v0.9.0 · 5326 in / 1330 out tokens · 32761 ms · 2026-05-10T15:16:25.001535+00:00 · methodology

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Reference graph

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