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A submartingale for the probability of avoiding the origin in one-point interaction ground-state diffusion: $d \in \{2,3\}$

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Pith's one-line read This paper proves that the ground-state diffusion with a one-point interaction hits the origin with positive probability in d=2 and d=3, and identifies the first hitting time as a truncated generalized inverse Gaussian law.

desk verdict Clean submartingale argument with a genuinely new d=3 result, but the d=3 SDE representation is imported from an unpublished preprint via a 'can be checked' step that must be completed before the GIG hitting law is fully established. read the letter →

arxiv 2608.11178 v1 pith:4NQHYS3O submitted 2026-08-11 math.PR

Barkat Mian This is my paper · ORCID
classification math.PR MSC 60J6060G4460J65
keywords ground-statediffusionone-pointinteractionsingularSchrödingeroperatorsubmartingalefirsthittingtimegeneralizedinverseGaussiandistributionDoobtransformmodifiedBesselfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the finite-horizon ground-state diffusion attached to a one-point attractive interaction in dimensions two and three, a process whose transition density is a Doob transform of the heat kernel of a Schr\"odinger operator with a delta potential at the origin. The paper constructs a bounded continuous submartingale whose increasing part grows only when the diffusion sits at the origin, and uses optional stopping to prove that the process hits the origin with positive probability on any time interval $[0,T]$. It identifies the first hitting time: conditionally on hitting by time $T$, the hitting time has the truncated generalized inverse Gaussian distribution $\mathrm{GIG}(1-d/2; |x|, \sqrt{2\lambda_\gamma})$, with tail $\mathbb{P}^{T,\gamma}_x[\tau>T] = K_{d/2-1}(\sqrt{2\lambda_\gamma}|x|, \lambda_\gamma T)/K_{d/2-1}(\sqrt{2\lambda_\gamma}|x|)$. It further shows that the process conditioned to avoid the origin is not a Brownian motion but carries a regularized time-dependent drift plus a continuous martingale. The point matters because ordinary Brownian motion in $d\ge 2$ never reaches the origin; the result shows the interaction reverses this in a precise, computable way.

What carries the argument

The load-bearing object is the ratio $S^\gamma_t(x)=e^{-\lambda_\gamma t}(g_t*\psi_\gamma)(x)/\psi_\gamma(x)$, which equals the probability that the diffusion avoids the origin up to time $t$. Time-reversing it along the process, $S^\gamma_{T-t}(X_t)$ is a bounded continuous submartingale whose Doob\textendash Meyer increasing part $A$ is flat on every excursion away from the origin and moves only at visits to the origin. The identity $\nabla S^\gamma_t = -S^\gamma_t(b^\gamma-\hat b^\gamma_t)$ connects the drift $b^\gamma=\nabla\log\psi_\gamma$ with the regularized drift $\hat b^\gamma_t=\nabla\log(g_t*\psi_\gamma)$, and the Gaussian-convolution lemma expresses $g_t*\psi_\gamma$ through the incomplete modified Bessel function $K_{d/2-1}(\cdot,\lambda_\gamma t)$, which converts $S$ into the explicit tail ratio in Corollary 1.5.

What would settle it

Compute the claimed tail probability for a concrete case, say $d=3$, $\gamma=1$, $x=(1,0,0)$, $T=1$, and compare it with a Monte Carlo simulation of the SDE (1.14) with drift $-(1+1/|X|)X/|X|$, run on paths that stay away from the origin; a mismatch would contradict the identity $\mathbb{P}^{T,\gamma}_x[\tau>T]=S^\gamma_T(x)$. Alternatively, verify directly whether the function $Q^\gamma_t$ of Appendix A.2 satisfies the three equations cited from [28, Eqs. (3.4)\textendash(3.6)]; failure of that check would break the local Brownian-motion representation used to keep $A$ constant off the origin.

Watch

Extended reading notes

Core claim

The central claim is that for $d\in\{2,3\}$ the finite-horizon ground-state diffusion hits the origin with positive probability, and the first hitting time $\tau$ obeys a truncated generalized inverse Gaussian law. On the event $\{\tau\le T\}$, the density is given by a ratio of the incomplete to the complete modified Bessel function, and the paper derives it in one stroke from the submartingale identity, rather than by solving a separate boundary-value problem. Under the conditional law $\mathbb{P}^{T,\gamma}_x[\,\cdot\,|\tau>T]$, the coordinate process is shown to satisfy $X_t = x + \int_0^t \hat b^\gamma_{T-s}(X_s)\,ds + \hat M_t^{T,\gamma}$ for a continuous martingale $\hat M^{T,\gamma}$, so the conditioning does not erase the drift entirely, in contrast to the total-mass diffusions. The same argument covers $d=2$ and $d=3$, with the dimension entering only through the order of the Bessel functions.

Load-bearing premise

The argument assumes the finite-horizon ground-state diffusion exists with transition density (1.9) and satisfies the SDE (1.14) away from the origin, and for $d=3$ it relies on a martingale-property check for the map $Q^\gamma_t$ that is asserted, not proved, by reference to prior work.

Editorial extensions

If this is right

  • The explicit tail $\mathbb{P}^{T,\gamma}_x[\tau>T] = K_{d/2-1}(\sqrt{2\lambda_\gamma}|x|,\lambda_\gamma T)/K_{d/2-1}(\sqrt{2\lambda_\gamma}|x|)$ holds for both $d=2$ and $d=3$ with $\lambda_\gamma=\gamma$ and $\lambda_\gamma=\gamma^2/2$ respectively.
  • Conditionally on $\{\tau\le T\}$, the first hitting time has the $\mathrm{GIG}(1-d/2;|x|,\sqrt{2\lambda_\gamma})$ density truncated to $(0,T]$, so the point potential produces a hitting-time law different from the total-mass diffusions.
  • Under conditioning to avoid the origin, the ground-state diffusion is not Brownian; it moves with the regularized drift $\hat b^\gamma_{T-t}$ plus a continuous martingale.
  • The submartingale's increasing part provides a pathwise marker of origin visits that can be used to study occupation-time and excursion properties of the ground-state diffusion.
  • Because the same submartingale construction works in $d=2$ and $d=3$, the paper offers a single mechanism for two cases that previously required separate arguments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same $S$-ratio construction should give explicit hitting probabilities for other Doob-transformed point-interaction diffusions whose driving families satisfy the heat equation away from the origin, since only that equation and the local SDE are used.
  • A natural next step would be to test numerically whether the conditioned ground-state diffusion in $d=3$ has the same large-scale transport behaviour as the conditioned total-mass diffusion, given that the latter loses all drift under conditioning.
  • The truncated-GIG form suggests a time-inversion symmetry: the hitting-time density is the same expression one would get from a Brownian bridge calculation, and one could try to extend the submartingale argument to occupation-time functionals such as $\int_0^T 1_{\{X_t\in B_\varepsilon\}}dt$.
  • Since the paper stops at $d=2,3$, the $S$-ratio approach might be the natural entry point for $d\ge4$, where the ground state is less singular at the origin, though the paper does not address that regime.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies the finite-horizon ground-state diffusion (GSD) associated with a one-point Schrödinger perturbation in dimensions d=2 and d=3. For the driving family Ψ_t^γ = e^{λγ t}ψ_γ, it defines S_t^γ(x) = e^{-λγ t}(g_t*ψ_γ)(x)/ψ_γ(x) and proves, via a bounded submartingale S_{T-t}^γ(X_t) whose increasing part grows only at visits to the origin, that the probability of avoiding the origin by time T equals a ratio of incomplete to complete modified Bessel K functions. It derives the conditional density of the first hitting time as a truncated generalized inverse Gaussian law and describes the dynamics under conditioning to avoid the origin, where a regularized drift remains. The main mathematical steps are the submartingale decomposition (Proposition 1.3), the square-integrability estimates of Section 3, and the optional stopping argument yielding Corollary 1.5.

Significance. If the proof gaps are filled, the paper gives a genuinely unified submartingale derivation of the survival probability and GIG hitting law for the GSD in d=2 and d=3, and it extends the total-mass-diffusion submartingale technique of [8] and [28] to the ground-state setting. The central formulas are explicit and parameter-free, the d=2 result agrees with the independent known GIG law in [6,15], and the conditioned dynamics are cleanly separated from the TMD case. The strength of the paper is its mechanism: S is an explicit ratio, the martingale part is identified in (1.17), and the comparison principle in Lemma 2.1 plus the integrability estimates in Lemma 3.1 are self-contained for both dimensions. The d=3 result is the new content, and its current dependence on an unverified import from an unpublished preprint is the main obstacle to accepting the paper as it stands.

major comments (3)
  1. [Appendix A.2, Lemma 1.2(ii)] The d=3 SDE representation is the load-bearing input for Proposition 1.3 and hence for Corollary 1.5, but its proof is not actually supplied. After defining Q^γ_t in (A.2), the text states twice that “it can be checked” that Q^γ_t satisfies [28, Eqs. (3.4)–(3.6)], and then imports [28, Lem. 3.3] and [28, Cor. 3.5] to obtain the Brownian motion W via (A.5). The missing check concerns precisely the generator identity and the behavior of ∇†Q^γ at zero that justify the martingale property and the quadratic variation; if the check fails, the local SDE (1.14) is unavailable, and the excursion-wise constancy of A in Proposition 1.3 is unproved. Please supply the full verification or replace this import by a direct self-contained calculation.
  2. [Appendix A.2, paragraph duplication] The paragraph beginning “Thus the function Q^γ_t(x) has the same eigenvalue decomposition…” appears twice verbatim. The first occurrence asserts that Y^{T,γ} is a continuous square-integrable P-martingale; the second asserts only that it is a continuous local martingale and then uses Lemma A.1 to prove square integrability. As printed, the proof asserts square integrability before establishing the local martingale structure. The duplication should be removed, and the logical order should be: local martingale and quadratic variation, then square integrability via Lemma A.1, then the Brownian motion W.
  3. [Corollary 1.5(ii) and Remark 1.6] The prefactor in the conditional density is typeset in a malformed way: it appears as (|x|√(2λ_γ))^{d/2−1} with an empty denominator in the fraction. The factor needed for consistency with the GIG parameterization in Remark 1.6, and also with the differentiation of the incomplete Bessel function in Appendix C, is (|x|/√(2λ_γ))^{d/2−1}. With the prefactor as printed, the normalization of the conditional density fails in d=3. Please correct the typesetting and verify the prefactor directly from the derivative leading to (C.3).
minor comments (3)
  1. [Appendix B, proof of Proposition 1.3] In the first display of the submartingale proof, the kernel denoted g^γ_{T-s}(X_s,y) is not defined; the argument appears to require p^γ_{t-s}(X_s,y), followed by the lower bound p^γ ≥ g_{t-s}. Please correct this typo.
  2. [Section 3.1, end of the regularized drift case] The sentence “Combining the above estimates with, we conclude…” contains a dangling “with” and should be rewritten; a full copyedit is also needed in several other places, including “The proof of following lemma” before Lemma 2.2.
  3. [General notation] The notation ˜b^γ_t(x) is introduced in (1.11) but in several places, including Remark 1.4 and Lemma 2.4, the regularized drift is written without the time subscript or with slightly different spacing; please make the notation uniform across the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the GIG hitting law is derived by optional stopping of an explicit submartingale, not fitted, and the d=2 result matches independent benchmarks; the unverified 'can be checked' import of the d=3 SDE from [28] is a correctness gap, not circularity.

full rationale

The derivation chain is self-contained in the sense needed for circularity analysis. The key quantity S^γ_t(x)=e^{-λγt}(g_t*ψγ)(x)/ψγ(x) is an explicit function of the free heat kernel and the ground state, not a fitted or renamed version of the target hitting probability. Proposition 1.3 obtains the submartingale property from the transition density (1.9), the positivity of h^γ in (1.3), the PDE (1.15), and the local SDE (1.14); optional stopping at τ∧T then yields P_x[τ>T]=S_T(x), and Lemma 2.2 evaluates this as a ratio of incomplete to complete Bessel functions. The d=2 GIG law is independently confirmed by [6, Eq. (7.5)] and [15, p. 884], providing an external benchmark. The d=3 case rests on the SDE representation imported from the author's own [28]: Appendix A.2 twice asserts 'it can be checked' that Q^γ_t satisfies [28, Eqs. (3.4)–(3.6)] and then borrows [28, Lem. 3.3] and [28, Cor. 3.5]. If that check fails, the excursion constancy of A and hence Corollary 1.5 in d=3 would be unsupported. This is a genuine verification gap, not a circular reduction: the target GIG density is nowhere assumed, no parameter is fitted to hitting data, and no equation is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central computation has no fitted constants. The model parameter gamma and the horizon T are inputs, and lambda_gamma equals gamma in d=2 and gamma^2/2 in d=3 by the ground-state eigenrelation. The load-bearing background consists of the existence of the diffusion and a few cited kernel bounds, listed above.

assumptions (4)
  • domain assumption There exists a unique finite-horizon Markov process with transition density q_t^gamma (Lemma 1.1), constructed in [6] for d=2 and in [12, Thm 3.3] for d=3.
    The whole paper is about this process; its existence is cited, not re-proved.
  • domain assumption For d=3, the coordinate process satisfies the SDE (1.14) locally away from the origin (Lemma 1.2).
    The proof in Appendix A.2 delegates the core martingale property to [28, Lem. 3.3] and says the needed identities 'can be checked', so the SDE is an imported result rather than a fully demonstrated one.
  • standard math Two-sided heat kernel bounds: (3.10) from [8, Prop. 8.17] for d=2 and (3.26) from [17, Lem. 8] for d=3.
    Used in Lemma 3.1 for square-integrability; cited from prior work.
  • standard math Uniform second-moment estimate for the 2d GSD from [6, Eq. (4.103)].
    Used in Lemma 1.7 to prove L2-boundedness of Z cM.

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Cite this review

Pith. "Pith review of A submartingale for the probability of avoiding the origin in one-point interaction ground-state diffusion: $d \in \{2,3\}$." pith.science (2026). https://pith.science/paper/4NQHYS3O

@misc{pith2026260811178,
  author       = {Pith},
  title        = {Pith review of: A submartingale for the probability of avoiding the origin in one-point interaction ground-state diffusion: $d \in \2,3\$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NQHYS3O}},
  note         = {Machine review of arXiv:2608.11178}
}
abstract

We study the near-origin behavior on $[0,T]$ of the singular diffusion whose transition density is given by a Doob transform of the integral kernel of the semigroup generated by the $d$-dimensional Schr\"odinger operator $L^{\gamma}$ with a one-point potential at the origin, where the driving family is the ground state of $L^{\gamma}$ and $d\in\{2,3\}$. We construct a submartingale whose increasing component grows only at times when the diffusion visits the origin. Using this submartingale, we show that the diffusion hits the origin with positive probability and that, conditionally on hitting the origin by time $T$, the first hitting time has a truncated generalized inverse Gaussian (GIG) distribution. We further study the dynamics under conditioning to avoid the origin: under the conditional law, the diffusion is not a standard Brownian motion, but instead admits a representation in terms of a regularized drift and a continuous martingale. While these properties are known in dimension two, the present submartingale-based approach provides an alternative verification and treats dimensions two and three in a unified manner.

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