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arxiv: 2604.23883 · v1 · submitted 2026-04-26 · 🧮 math.PR

Sharp pathwise nonuniqueness for additive SDEs

Elias Hess-Childs , Keefer Rowan This is my paper

Pith reviewed 2026-05-08 05:19 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic differential equationspathwise uniquenessweak uniquenessHölder continuous driftadditive noiseBerry-Esseen theoremfractional Brownian motion
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The pith

For any negative Hölder exponent there exists a velocity field giving unique weak SDE solutions but no pathwise uniqueness.

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

The paper shows that the classical threshold where additive Brownian noise guarantees strong well-posedness for SDEs is sharp. For any α less than zero it constructs a bounded velocity field of Hölder regularity α that admits a unique weak solution yet fails pathwise uniqueness. The construction is random and uses central-limit scaling together with the Berry-Esseen theorem. This means that below zero regularity Brownian noise is not enough to force unique strong solutions in general. The argument also extends to show nonuniqueness for other driving noises with suitable regularity and an analogous sharpness result for fractional Brownian motion.

Core claim

For every α < 0 the authors produce a random velocity field u in L^∞_t C^α_x almost surely such that the additive SDE with drift u and Brownian noise has a unique weak solution but does not satisfy pathwise uniqueness and therefore possesses no strong solutions. This stands in contrast to the regime α ≥ 0 where Zvonkin-Veretennikov-Davie theory already guarantees a unique strong solution. The field is built by central-limit scaling of a suitable base function, with the Berry-Esseen theorem controlling the approximation so that the Hölder condition holds almost surely while pathwise uniqueness is destroyed.

What carries the argument

A randomly scaled velocity field obtained by central-limit averaging that remains in the Hölder space C^α almost surely while eliminating pathwise uniqueness.

Load-bearing premise

The randomly constructed velocity field lies in the Hölder space C^α almost surely.

What would settle it

A proof that every velocity field in L^∞_t C^α_x for α < 0 yields pathwise uniqueness for the corresponding SDE would show no such counterexample exists.

read the original abstract

We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin--Veretennikov--Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift $u$ and show that for any $\alpha<0$, there exists a velocity field $u \in L^\infty_t C^\alpha_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $\alpha \geq 0$, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses central limit theorem scaling through the Berry--Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain H\"older regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions.

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 constructs, for every α < 0, a random velocity field u ∈ L^∞_t C^α_x such that the additive SDE dX_t = u(t, X_t) dt + dB_t admits a unique weak solution but fails pathwise uniqueness (and therefore has no strong solutions). The construction proceeds by random central-limit scaling of a base field, with the Berry–Esseen theorem supplying quantitative approximation rates; extensions to non-Brownian noises and fractional Brownian motion are also given. This is presented as a sharpness result for the Zvonkin–Veretennikov–Davie regime of strong well-posedness by noise.

Significance. If the construction is fully rigorous, the result supplies explicit counterexamples showing that the Hölder threshold α ≥ 0 is optimal for pathwise uniqueness in additive SDEs. The random-field approach and the quantitative use of Berry–Esseen are technically interesting, and the extensions to other driving noises broaden the scope. The work would clarify the boundary between regularization by noise and the onset of non-uniqueness.

major comments (1)
  1. [§3] §3 (random-field construction via CLT scaling): The claim that the resulting random field belongs to C^α almost surely for every α < 0 rests on Berry–Esseen error bounds. These bounds control finite-dimensional or mollified marginals at fixed scales, but the argument does not explicitly verify that the supremum of the Hölder seminorm (taken over all scales and test functions) remains finite with probability one. Without uniform tail estimates that prevent logarithmic divergence across dyadic scales, the a.s. membership in C^α is not yet secured; this is load-bearing for the existence statement.
minor comments (1)
  1. [Abstract] The abstract states that the field 'admits a unique weak solution'; a brief indication of how weak uniqueness is established (e.g., via Girsanov or martingale-problem arguments) would help the reader before the pathwise-non-uniqueness part is presented.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to strengthen the exposition of the almost-sure Hölder regularity in the random-field construction. We respond to the single major comment below.

read point-by-point responses

  1. Referee: The claim that the resulting random field belongs to C^α almost surely for every α < 0 rests on Berry–Esseen error bounds. These bounds control finite-dimensional or mollified marginals at fixed scales, but the argument does not explicitly verify that the supremum of the Hölder seminorm (taken over all scales and test functions) remains finite with probability one. Without uniform tail estimates that prevent logarithmic divergence across dyadic scales, the a.s. membership in C^α is not yet secured; this is load-bearing for the existence statement.

    Authors: We appreciate the referee pointing out that the passage from scale-by-scale Berry–Esseen control to a uniform almost-sure bound on the Hölder seminorm requires more explicit justification. The construction in §3 applies the quantitative Berry–Esseen rates at each fixed dyadic scale to obtain moment bounds on the increments of the rescaled field. These rates are strong enough (polynomial decay in the scale, uniform in the test functions) that a standard Borel–Cantelli argument over the countable collection of dyadic scales yields summable probabilities for the event that the seminorm at scale 2^{-k} exceeds a threshold growing like k^2. This choice of threshold rules out logarithmic blow-up and ensures the supremum over all scales is finite with probability one. While the manuscript invokes this idea briefly, we agree that the chaining and tail estimates should be written out in full. We will add a short lemma in §3 that records the uniform tail bound and the resulting almost-sure conclusion. revision: yes

Circularity Check

0 steps flagged

No circularity; construction uses external probabilistic tools

full rationale

The derivation constructs a random velocity field via standard central-limit scaling and the Berry-Esseen theorem to produce a counterexample in L^∞_t C^α_x for α<0. This relies on external theorems rather than self-citations, fitted parameters, or redefinitions. No step reduces the claimed nonuniqueness or Hölder regularity to an input by construction; the argument remains self-contained against the classical Zvonkin-Veretennikov-Davie regime.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard axioms of Brownian motion and Hölder spaces; no free parameters or new entities.

axioms (2)
  • standard math Existence and basic properties of Brownian motion on R^d
    Invoked for the SDE dX = u dt + dB
  • standard math Berry-Esseen quantitative central limit theorem
    Controls the random field scaling

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