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arxiv: 2606.08008 · v1 · pith:A7K6O4IQnew · submitted 2026-06-06 · 🧮 math.PR

Pathwise structure of the three-dimensional attractive one-point interaction diffusion

Barkat Mian This is my paper

Pith reviewed 2026-06-27 19:36 UTC · model grok-4.3

classification 🧮 math.PR
keywords one-point interaction diffusionattractive delta potentialthree-dimensional Brownian motionpathwise behaviorDoob-Meyer decompositionlocal SDEvisiting the origin
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The pith

The three-dimensional attractive one-point interaction diffusion visits the origin with positive probability and its avoidance-conditioned law is Wiener measure.

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

The paper studies the paths of the diffusion whose law was constructed earlier for the attractive delta interaction at a point in three dimensions. It derives a local stochastic differential equation obeyed by the process when it is away from the origin and uses that equation to build a submartingale. The compensator of this submartingale is supported exactly on the times the process is at the origin. From the existence of the submartingale it follows that the process reaches the origin with positive probability. Conditioning the entire law on the event of never hitting the origin then yields ordinary three-dimensional Brownian motion.

Core claim

The process satisfies a local SDE away from the origin. A natural submartingale is constructed from this SDE; its Doob-Meyer increasing component is supported on the set of times at which the process visits the origin. This implies that the process visits the origin with positive probability. The law of the process conditioned on avoiding the origin coincides with three-dimensional Wiener measure.

What carries the argument

A submartingale built from the local SDE satisfied away from the origin, whose Doob-Meyer increasing process is carried by visits to the origin.

If this is right

  • The process reaches the origin with positive probability.
  • Conditioning on the event of never visiting the origin recovers exactly three-dimensional Brownian motion.
  • The compensator of the constructed submartingale is supported only on origin visits.
  • The path measure is singular with respect to Wiener measure because of the positive probability of hitting the origin.

Where Pith is reading between the lines

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

  • The result distinguishes the attractive case from the repulsive one-point interaction, where avoidance might hold with probability one.
  • Numerical path simulations could estimate the hitting probability as a function of beta to check consistency with the submartingale construction.
  • The same submartingale technique might extend to other singular potentials whose local behavior away from the singularity is known.

Load-bearing premise

The diffusion under study is exactly the one previously constructed for the Hamiltonian with positive beta and obeys the identified local SDE away from the origin.

What would settle it

A direct computation or simulation showing that paths avoid the origin almost surely under this law would falsify the positive visiting probability.

read the original abstract

We study the pathwise behavior of the three-dimensional attractive one-point interaction diffusion whose law was constructed by Cranston, Koralov, Molchanov and Vainberg, corresponding to the singular Schr\"odinger Hamiltonian \[ \frac12\Delta+\frac{\beta}{2}\delta_0, \qquad \beta>0. \] We identify a local stochastic differential equation satisfied by the process away from the origin and use it to construct a natural submartingale whose increasing component in the Doob-Meyer decomposition is supported on the set of times at which the process visits the origin. In particular, we show that the process visits the origin with positive probability and that the law conditioned on avoiding the origin is three-dimensional Wiener measure.

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 / 0 minor

Summary. The paper studies the pathwise behavior of the three-dimensional attractive one-point interaction diffusion whose law was constructed by Cranston, Koralov, Molchanov and Vainberg for the Hamiltonian ½Δ + (β/2)δ₀ with β>0. It identifies a local SDE satisfied by the process away from the origin and constructs a submartingale whose Doob-Meyer increasing component is supported exactly on the times at which the process visits the origin. This yields the claims that the process visits the origin with positive probability and that its law conditioned on avoiding the origin coincides with three-dimensional Wiener measure.

Significance. If the central claims are established rigorously, the work supplies a precise pathwise description of the role played by the singularity in this class of diffusions with point interactions. The identification of the conditioned process as Wiener measure is a clean and potentially useful structural result that connects singular Schrödinger operators to standard Brownian motion.

major comments (1)
  1. [submartingale construction following local SDE identification] The construction of the submartingale (described in the abstract and developed after the local SDE identification) applies an Itô-type formula to a test function that is singular at the origin. The manuscript must supply an explicit justification—via a local-time correction, Tanaka-Meyer formula, or direct verification of the finite-variation terms—that no extraneous compensator arises off the origin; without this, the support claim for the increasing process, and therefore both the positive-probability visit conclusion and the conditional Wiener-measure statement, do not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point below and will incorporate the requested justification in the revision.

read point-by-point responses

  1. Referee: The construction of the submartingale (described in the abstract and developed after the local SDE identification) applies an Itô-type formula to a test function that is singular at the origin. The manuscript must supply an explicit justification—via a local-time correction, Tanaka-Meyer formula, or direct verification of the finite-variation terms—that no extraneous compensator arises off the origin; without this, the support claim for the increasing process, and therefore both the positive-probability visit conclusion and the conditional Wiener-measure statement, do not follow.

    Authors: We agree that an explicit justification is required for the application of the Itô-type formula to the singular test function. The current manuscript relies on the local SDE identification to control the behavior away from the origin but does not provide a separate verification that the finite-variation terms introduce no extraneous compensator off the origin. In the revised version we will add a dedicated paragraph (or short subsection) that applies the Tanaka-Meyer formula directly to the chosen test function and verifies that all finite-variation contributions are supported exclusively on the set of origin visits. This addition will make the support claim for the Doob-Meyer increasing process fully rigorous and thereby secure both the positive-probability visit statement and the conditional Wiener-measure identification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new path properties derived from external prior construction

full rationale

The paper explicitly takes the process law as constructed by Cranston, Koralov, Molchanov and Vainberg as given (different authors, external reference). It then identifies a local SDE away from the origin and constructs a submartingale whose Doob-Meyer compensator is supported on origin visits. These steps use the given law to derive the positive-probability visit claim and the conditional Wiener measure without any reduction of the new claims to fitted parameters, self-definitions, or self-citation chains. No equations or arguments in the provided text exhibit the enumerated circularity patterns; the derivation remains self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior existence result for the process and on the identification of its local dynamics away from zero; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The process is the diffusion whose law was constructed by Cranston, Koralov, Molchanov and Vainberg for the Hamiltonian ½Δ + (β/2)δ₀, β>0
    The paper takes this construction as the starting point for its pathwise analysis.

pith-pipeline@v0.9.1-grok · 5642 in / 1151 out tokens · 21858 ms · 2026-06-27T19:36:33.627539+00:00 · methodology

discussion (0)

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Reference graph

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