{"id":"ef25b695-f67a-4e2a-854a-1c61de0ff307","arxiv_id":"2608.11178","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the ground-state one-point interaction diffusion in dimensions 2 and 3, the probability of avoiding the origin up to time T equals a ratio of incomplete Bessel functions, and the first hitting time is generalized inverse Gaussian.","lead":"This paper derives the exact probability that a singular diffusion with an attractive point potential at the origin avoids the origin up to time T, and finds the distribution of the first hitting time. The result is a generalized inverse Gaussian law, with the same formula in dimensions two and three.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The d=3 SDE representation is imported from [28] via an unverified 'can be checked' assertion; if [28, Eqs. (3.4)–(3.6)] fail for the GSD's Q function, Proposition 1.3's excursion argument and Corollary 1.5 collapse in d=3.","rationale":"I read the paper in good faith. The d=2 argument is coherent, the submartingale construction is elegant, and the closed-form survival probability follows once the SDE representation and the excursion-constancy of A are granted. The d=3 case, however, depends entirely on Lemma 1.2(ii), whose proof in Appendix A.2 is not self-contained: it says that Q^γ_t 'can be checked' to satisfy [28, Eqs. (3.4)–(3.6)] and then borrows [28, Lem. 3.3] and [28, Cor. 3.5]. Since [28] is an unpublished preprint and the check is not performed, this is the load-bearing gap. The duplicated passage in Appendix A.2, where Y is first claimed to be a square-integrable martingale and then derived as a local martingale followed by an L^2 bound, reinforces that the d=3 proof was not fully written out. I also note a separate textual error: the unified eigenfunction formula (1.5) and Lemma 2.2 use |x|^{d/2-1} in the d=3 case, whereas the proof in Appendix A.3 and the explicit formula (1.8) imply the exponent should be 1-d/2; this cancels in the ratio defining S^γ_T(x), so it does not by itself invalidate Corollary 1.5, but it should be corrected. Given the reader's CONDITIONAL verdict already targets the d=3 SDE import, I see no reason to move the verdict; completing the d=3 check and fixing the sign typo would move it toward ACCEPT.","tokens_in":43335,"tokens_out":25440,"duration_ms":217184,"concrete_test":"Independently verify for d=3 that Q^γ_t of (A.2) satisfies [28, Eqs. (3.4)–(3.6)]: (i) compute E^γ_x[Q^γ_{T-t}(X_t)] with q^γ_t from (1.9) and check it equals Q^γ_T(x); (ii) compute the quadratic variation of u·Y from ∇†Q^γ_{T-t} and check it equals ∫ |∇†Q^γ_{T-s}(X_s)u|^2 ds; (iii) verify that the inverse-jacobian integral (A.5) has the Lévy character of a Brownian motion. If (i)–(iii) fail, Lemma 1.2(ii) is false and Corollary 1.5(d=3) has no proof. A direct re-derivation should replace the 'can be checked' step in Appendix A.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula for d=3 rests on Lemma 1.2(ii), the local SDE representation for X. In Appendix A.2 this is not proved: 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 a Brownian motion W via (A.5). Proposition 1.3 then uses (1.14) on excursion intervals [a,b] to show dA=0 there; Corollary 1.5 uses A_{t∧τ}=0 to identify E[S_{τ∧T}]=P_x[τ>T]. If the check fails—for instance, if the generator identity behind [28, Eqs. (3.4)–(3.6)] acquires an extra term from the GSD's time-dependent Q or from boundary behavior of ∇†Q at zero—the Brownian W and the SDE do not exist in d=3, the excursion constancy of A is unproved, and the GIG hitting law for d=3 is not established. The paper itself flags this by importing an unpublished preprint and by leaving the check to the reader; the duplicated passage around Y, first claiming square-integrability and then deriving it from locality, does not remove the gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":43623,"tokens_out":17854,"duration_ms":146826,"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":[{"comment":"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.","section":"Appendix A.2, Lemma 1.2(ii)"},{"comment":"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.","section":"Appendix A.2, paragraph duplication"},{"comment":"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).","section":"Corollary 1.5(ii) and Remark 1.6"}],"minor_comments":[{"comment":"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.","section":"Appendix B, proof of Proposition 1.3"},{"comment":"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.","section":"Section 3.1, end of the regularized drift case"},{"comment":"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.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The main d=3 technical step is imported from [28], which is the author's own unpublished preprint, and the verification of the imported conditions is left to the reader. This is a concern for the journal's standards of self-containedness. The duplicate paragraph and the garbled prefactor in Corollary 1.5 also suggest that the manuscript is not yet in polished form. I see no issue of circularity: the survival formula is not fitted and the d=2 comparison with known results is a useful check. The paper is within scope for a probability journal, and the central idea is sound, but it should not be accepted until the d=3 SDE proof is either fully carried out or replaced by a direct derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the paper does something real and something incomplete. The 2d part is known, and the authors say so; the genuinely new content is the 3d hitting probability and conditioned dynamics, plus the unified treatment through an explicit submartingale. The central optional-stopping argument is clean: the function S_t^gamma is an explicit ratio, the submartingale's increasing component is constant off excursions, and the GIG tail and density follow directly. The d=2 formula matches prior results, which is good evidence that the mechanism is right.\n\nThe weakest point is exactly where the reader and stress-test put it. Appendix A.2 does not prove that Q^gamma_t satisfies the hypotheses [28, Eqs. (3.4)-(3.6)]; it says 'it can be checked' and then imports the martingale property and the Brownian motion from [28]. That check is load-bearing for d=3. If it fails, or if the generator identity acquires an extra term from the time-dependent Q, then the local SDE (1.14) is unavailable on excursions, the constancy of A on those excursions is unproved, and Corollary 1.5 in d=3 is not established. The paper itself flags the reliance on the unpublished preprint, and the verbatim duplicated paragraph around Y in A.2 is a sloppy copy-paste that makes the gap harder to ignore. None of this makes the result likely false; the ingredients are plausible and the d=2 check passes. But the d=3 proof is not self-contained and should not be treated as complete.\n\nThe long square-integrability estimates in Section 3 lean on cited bounds from [8] and [17]; that is normal practice, and the computations look careful. I did not find fitted parameters or circular reasoning. The literature is cited honestly, including the known 2d result.\n\nWho is this for? Researchers working on singular diffusions, point interactions, and polymer limits. A serious referee should see it, but the author needs to fill the d=3 gap and clean up the duplication before publication. I would send it to peer review with that expectation.","headline":"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.","tokens_in":44196,"tokens_out":1670,"would_cite":false,"duration_ms":17428,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J60","60G44","60J65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["ground-state diffusion","one-point interaction","singular Schrödinger operator","submartingale","first hitting time","generalized inverse Gaussian distribution","Doob transform","modified Bessel functions"],"falsifier":"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.","tokens_in":43096,"feed_emoji":"🎯","tokens_out":9167,"duration_ms":74205,"temperature":0.7,"pith_summary":"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.","feed_headline":"Ground-state diffusion hits origin in d=2 and d=3","feed_subtitle":"A submartingale whose growth marks origin visits yields the hitting probability and a generalized inverse Gaussian law.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Constructs the two-dimensional ground-state diffusion and its SDE representation; supplies the existence input and the d=2 SDE used in Lemmas 1.1 and 1.2.","marker":"[6]"},{"why":"Constructs the three-dimensional ground-state diffusion and identifies its generator; the finite-horizon law in Lemma 1.1 comes from this construction.","marker":"[12]"},{"why":"Establishes the analogous submartingale for the 3d total-mass diffusion and the local SDE in three dimensions; the proof of Lemma 1.2 imports its martingale-property check for the map Q.","marker":"[28]"},{"why":"Gives the two-dimensional SDE representation (Prop. 4.2(4)) that the paper converts into the present d=2 drift form.","marker":"[7]"},{"why":"Provides the submartingale framework for the 2d total-mass diffusion and the kernel bound used in the d=2 integrability estimate.","marker":"[8]"},{"why":"Supplies the two-sided kernel bounds for the 3d point-interaction heat kernel used in Lemma 3.1.","marker":"[17]"},{"why":"Records the known two-dimensional GIG hitting-time law, which the paper's Corollary 1.5 rederives.","marker":"[15]"}],"fun_headline_variants":["Submartingale reveals origin-hitting law in d=2,3","Ground-state diffusion: hitting time has truncated GIG law","Conditioned ground-state diffusion gains drift, not Brownian","Unified submartingale proof for origin hits in d=2,3","Origin hits from submartingale: GIG distribution in d=2,3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Submartingale reveals origin-hitting law in d=2,3","Ground-state diffusion: hitting time has truncated GIG law","Conditioned ground-state diffusion gains drift, not Brownian","Unified submartingale proof for origin hits in d=2,3","Origin hits from submartingale: GIG distribution in d=2,3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000731,"raw_usage":{"total_tokens":3289,"prompt_tokens":980,"completion_tokens":2309,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":2216}},"tokens_in":596,"tokens_out":2309,"duration_ms":14891,"temperature":1.0,"reasoning_tokens":2216,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:36:15.359378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Pathwise structure of the three-dimensional attractive one-point interaction diffusion","cited_arxiv_id":"2606.08008","evidence_quote":"Establishes the analogous submartingale for the 3d total-mass diffusion and the local SDE in three dimensions; the proof of Lemma 1.2 imports its martingale-property check for the map Q."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Records the known two-dimensional GIG hitting-time law, which the paper's Corollary 1.5 rederives."}],"review_version":1}