{"id":"c6adafe2-eacc-445d-8d09-91737bcdb1d5","arxiv_id":"2505.01703","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the 2D N-body delta-Bose gas with all but one delta interaction switched off, explicit transition densities, invariant measure, Harris recurrence and singular-drift SDEs are proved for the one-delta stochastic motion.","lead":"This paper builds and analyzes random motions for a two-dimensional many-body quantum gas in which only one pair of particles interacts. It produces explicit probability formulas and differential equations for those motions, which are meant to be the foundation for the full many-body construction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The singular drift in the stated SDEs (1.21), (1.22), (4.1), (4.3), and (4.15) uses 1/Z instead of 1/\\bar{Z}; the imaginary-component drift has the wrong sign, contradicting the radial BES(0,β↓) law and the paper's own generator (1.18).","rationale":"The reader's weakest assumption concerned the imported BES(0,β↓) machinery from [11] and [18]. That is a reasonable external-dependency concern, but the paper's proofs and formulas show a more immediate internal problem: the stated SDEs and the generator (4.13) use the wrong complex reciprocal, 1/Z instead of 1/\\bar{Z}. This is not merely a notational preference: the real drift vector differs in the sign of the imaginary component, so the radial process would not be BES(0,β↓) if the SDE were taken literally, and the generator would not match (1.18). The paper's own derived equations, especially (4.4)–(4.5) and the proof of Proposition 5.3, use the corrected form, indicating that the intended theorem is recoverable. Therefore the appropriate verdict is CONDITIONAL: the central SDE claim should be accepted only after the displayed SDEs and (4.13) are corrected to use 1/\\bar{Z} (equivalently Z/|Z|²) and z1/|z1|², respectively, and after verifying that all subsequent applications in [12,13] use the corrected form. The existence, marginal formulas, invariant measure, and Harris recurrence results may still stand, but the SDE statement as written is false, so an unconditional accept is not warranted.","tokens_in":62598,"tokens_out":33401,"duration_ms":285998,"concrete_test":"Re-derive (1.22) from (1.10) and (1.12) by Itô's formula: compute dZ_t for Z_t = ρ_t exp(iγ_{∫_0^t ds/ρ_s²}) and ρ_t satisfying (1.12). The drift should be −√(2β)(K1/K0)(Z/|Z|) dt, which equals −(bK1/K0)(1/\\bar{Z}) dt. Equivalently, compute the infinitesimal generator of the stated SDE (1.22) at z = 1+i and compare with the generator (1.18) (after scaling by √2): the imaginary-component drift will have the opposite sign, settling whether 1/Z is a typo for 1/\\bar{Z}.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.1(3◦), Theorem 4.1(1◦), and Proposition 4.2(4◦) state the one-delta SDE with drift −(bK1(√2β|Z|)/K0(√2β|Z|))(1/Z) dt. The correct complex reciprocal is 1/\\bar{Z}, not 1/Z. From the skew-product representation (1.10) and the radial SDE (1.12), Itô's formula gives dZ = −√(2β)(K1/K0)(Z/|Z|) dt + dW = −(bK1/K0)(1/\\bar{Z}) dt + dW; the real drift vector is −a(x/r², y/r²) with a = bK1/K0. The stated 1/Z gives the vector −a(x/r², −y/r²), reversing the sign of the imaginary-component drift. For example, at z0 = i, the stated SDE gives an initial upward drift, while (1.18) and radial symmetry require a downward drift. The same error appears in the generator (4.13), which should have z1/|z1|² inside the inner product, not z1. The paper's own later formulas (4.4)–(4.5) and the derivation of (5.4) in Proposition 5.3 are consistent with the corrected 1/\\bar{Z} form, so the displayed SDEs are internally inconsistent with the process they are claimed to govern. Since the SDE is a central claim of Theorem 1.1 and is used in the follow-up papers, this is a load-bearing defect as printed.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper is the first in a series on stochastic motions for the two-dimensional N-body delta-Bose gas. It studies the stochastic one-delta motion, a C^N-valued diffusion obtained by coupling a two-dimensional Bessel-type relative motion (BES(0,β↓)) with independent free Brownian particles. The main results are: explicit one-dimensional marginals and an explicit invariant measure (Theorem 2.1), Harris recurrence of the joint process with an independent Brownian motion (Theorem 3.1), and Langevin-type SDEs with singular drift for the interacting coordinate (Theorem 4.1 and Proposition 4.2). Section 5 provides a general framework for transforming radial/angular skew-product diffusions into complex SDEs. The paper is written as a foundation for later papers constructing many-delta motions and Feynman-Kac-type formulas.","tokens_in":62976,"tokens_out":19975,"duration_ms":177817,"significance":"If the corrections below are made, this is a substantial contribution. The paper contains explicitly computable transition densities and invariant measures for a singular diffusion associated with the one-delta specialization of the 2D delta-Bose gas, careful analytic estimates for the singularities (sharp negative moments with logarithmic corrections, continuity of weak-integrable convolutions, and control of differentiation near zeros), and a Harris recurrence result that is likely to be a key tool for the series. The proofs are detailed and generally careful; the paper also gives explicit formulas that can be checked directly. However, a systematic conjugation/sign error in the displayed complex SDEs and in the generator, together with an inconsistency in the definitional equation (1.19), are load-bearing as printed; these issues must be fixed before the central SDE theorem can be accepted.","major_comments":[{"comment":"The complex drift is displayed with 1/Z, but Itô's formula applied to the skew-product representation (1.10) and the radial SDE (1.12) yields 1/\\bar{Z}. Concretely, the radial drift term −√(2β)(K1/K0)(Z/|Z|) equals −(bK1/K0)(1/\\bar{Z}), not −(bK1/K0)(1/Z); the latter reverses the sign of the imaginary-component drift. For example, at z0 = i the displayed SDE gives an initial upward drift, while the generator (1.18) and radial symmetry require a downward drift. The paper's own equations (4.4)–(4.5) and the intermediate formula (5.7) are consistent with 1/\\bar{Z}, so the displayed SDEs are internally inconsistent. This is load-bearing because the SDE is a central claim and is used in the follow-up papers [12]–[14].","section":"Section 1.3 and Section 4, Eqs. (1.21), (1.22), (4.1), (4.3), (4.15); Section 5, Eq. (5.4)"},{"comment":"The generator A defined in (4.13) is not equivalent to (1.18) as stated. Since bK1(√(2β)|z1|) = √(2β)|z1|K1(√(2β)|z1|), the drift term in (4.13) is −√(2β)|z1|^2(K1/K0)⟨z1/|z1|, ∇f⟩, which has an extra |z1|^2 factor and the wrong sign relative to the drift of {Z_t}. The correct term should be −√(2β)(K1/K0)⟨z1/|z1|, ∇f⟩. Because Proposition 4.2(2◦)–(3◦) and hence the proof of the SDE in (4.15) rely on this generator, the displayed formula must be corrected for the proof to be valid.","section":"Section 4.2, Eq. (4.13) and Remark 4.3(1◦)"},{"comment":"As printed, the definition of the one-delta motion is not self-consistent. Substituting (1.19) into the definition Z^i_t = (Z^{i'}_t − Z^i_t)/√2 gives Z^i_t = z^i_0 + W^i_t + Z^i_t, where the Z^i_t on the right is the prescribed relative-motion process; this would force W^i_t = −z^i_0 for all t, contradicting that W^i is a standard Brownian motion. The construction needs a corrected formula, or an explicit statement that (1.19) is a representation derived from the SDE rather than a definition.","section":"Section 1.3, Eqs. (1.19) and (1.23)"}],"minor_comments":[{"comment":"The word 'bivarite' should be 'bivariate'.","section":"Section 1.3, Theorem 1.1(2◦)"},{"comment":"'two-dimensional standard Bronwian motion' should be 'Brownian motion'.","section":"Section 1.3, paragraph after Eq. (1.22)"},{"comment":"Remark 4.3(1◦) states that (4.13) is equivalent to (1.18); this is incorrect for the reason given in the major comment on Eq. (4.13), and the remark should be updated once the generator is corrected.","section":"Section 4.2, Remark 4.3(1◦)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this paper is the real thing: the analytic transition densities, local-time joint laws, Feller property with explicit invariant measure, Harris recurrence of the bivariate process, and the sharp negative moments with log corrections are substantial and, as far as I can tell, new. Second, the central SDE claim as printed is wrong. Theorem 1.1(3°), Theorem 4.1(1°), Proposition 4.2(4°), and (4.15) all state the drift with 1/Z_s. The correct factor is 1/\\bar{Z}_s (equivalently, in the generator (4.13), the vector in the inner product should be z/|z|², not z). The radial BES(0,β↓) law and the paper's own skew-product representation force the imaginary-part drift to have the opposite sign from what the displayed 1/Z gives. At z0 = i, the stated SDE pushes the imaginary component upward; the correct dynamics push it downward.\n\nThe telling evidence is internal: equations (4.4)-(4.5) for the squared and radial processes use Re(Z^j/Z^i) with the coefficient bK1/K0, which is what you get from the 1/\\bar{Z} drift, not from the 1/Z drift. The derivation of (5.4) in Proposition 5.3 also produces e^{iϑ}dAϱ, i.e., the corrected form. So this looks like a systematic typo repeated through the SDE sections, not a conceptual failure. But as it stands, Theorem 1.1(3°) is false, and the paper cannot be used for pathwise dynamics until every occurrence of 1/Z in the SDE statements and in (4.13) is changed.\n\nWhat the paper does well beyond that: the proofs are detailed and the difficult analysis items are handled honestly. The continuity of convolutions with weakly integrable kernels (Lemma 4.8) is a nice tool. The paper is honest about its scope: it is the degenerate one-delta case, and the many-body payoff sits in [12-14]. The dependence on the BES(0,β↓) machinery from [11] is real but it is a cited external construction, not a circularity.\n\nThe reader's soundness of 8 is too generous given the SDE statements are wrong. A corrected version would deserve a much higher score. My own take: this is a salvageable and important stepping stone, but the author needs to fix the SDE statements and the generator before the paper is used.\n\nRecommendation: Send it to a serious referee. The fix is straightforward but must be traced through Section 4, especially the proof of Proposition 4.7 and the continuity argument. I would not cite it in its present form; I would cite the corrected version.","headline":"Rich technical paper with a systematic sign error in the stated SDEs: the drift should use 1/\\bar{Z}, not 1/Z, and the same mistake infects the generator (4.13); the rest of the analysis looks solid and worth a serious referee after correction.","tokens_in":63491,"tokens_out":10819,"would_cite":false,"duration_ms":96379,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J55","60J65","60H30","81S40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes explicit probability laws, an invariant measure, and a singular-drift SDE for the stochastic one-delta motions of the two-dimensional many-body delta-Bose gas.","keywords":["delta-Bose gas","stochastic one-delta motions","Bessel process BES(0,beta downarrow)","skew-product diffusion","singular drift SDE","Macdonald functions","Harris recurrence","Feynman-Kac formula"],"falsifier":"Evaluate the local time at zero of the SDE (4.15) started at the origin and compare its Laplace transform with the paper's identity $\\mathbb{E}^{\\beta\\downarrow}_0[\\int_0^\\infty e^{-q\\tau}\\,dL_\\tau]=1/\\log(1+q/\\beta)$ for multiple $q$; equivalently, simulate the SDE and check its one-time marginals against the closed-form density (2.5). A mismatch at any fixed time would directly contradict Theorem 2.1 and the SDE characterization of Proposition 4.2.","tokens_in":62399,"feed_emoji":"🎲","tokens_out":7994,"duration_ms":84732,"temperature":0.7,"pith_summary":"The two-dimensional delta-Bose gas is a contact-interaction quantum many-body model whose pathwise stochastic description has been missing because delta potentials are invisible to planar Brownian motion. This paper constructs the stochastic one-delta motions: N particles in the plane where only one pair feels the contact interaction, carried by a special Bessel-type diffusion while all other particles move as free Brownian motions. It proves that these motions are Feller processes with explicit one-time marginal densities and an explicit invariant distribution, that the joint process with an independent Brownian coordinate is Harris recurrent, and that the paths satisfy a Langevin SDE whose drift is singular at collision yet well defined through the zero set. These are the building blocks the author plans to assemble into stochastic motions for the full N-body gas, so the paper's value is that it makes the singular-contact picture tractable at the level of individual paths.","feed_headline":"One-delta Bose motions: explicit laws and a singular SDE","feed_subtitle":"The interacting pair is a special diffusion until collision; these laws make the full many-body construction tractable.","key_machinery":"The central objects are the skew-product diffusion structure and the special radial process $\\mathrm{BES}(0,\\beta\\downarrow)$, the $\\mathbb{R}_+$-valued diffusion whose inverse local time at $0$ is a gamma subordinator and whose radial SDE is $\\rho_t=\\rho_0+\\int_0^t(\\frac{1}{2\\rho_s}-\\sqrt{2\\beta}\\frac{K_1(\\sqrt{2\\beta}\\rho_s)}{K_0(\\sqrt{2\\beta}\\rho_s)})\\,ds+B_t$. The angular part is fixed by the same skew-product representation used for planar Brownian motion: $Z_t=|Z_t|\\exp(i\\gamma_{\\int_0^t ds/|Z_s|^2})$, with an independent circular Brownian motion $\\gamma$, which determines the law up to the first hit of zero and is extended by conditioning. The paper's analytical engine consists of explicit integral identities for Macdonald functions, sharp negative-moment estimates with logarithmic corrections (Proposition 4.5), and a Kolmogorov forward equation (Proposition 4.2(3)) that is used to derive the SDE across the zero set. These tools convert the formally infinite delta interaction into a well-defined pathwise drift.","core_discovery":"For every interacting pair $i=(i',i)$ and coupling $\\beta_i>0$, the stochastic one-$\\delta$ motion is the $\\mathbb{C}^N$-valued diffusion obtained by taking the interacting coordinate $Z^i_t$ to be the relative-motion diffusion of the companion paper [11]—a skew-product diffusion whose radial part is the special Bessel-type process $\\mathrm{BES}(0,\\beta_i\\downarrow)$—and by adding independent planar Brownian motions for all other particles. The main theorems establish three supporting facts. First, the one-time marginals are explicitly given by a continuous transition density $p^{\\beta\\downarrow}_t(z_0,z_1)$ and an invariant probability measure $\\mu_0^{\\beta\\downarrow}(dz)=\\frac{2\\beta}{\\pi}K_0(\\sqrt{2\\beta}|z|)^2\\,dz$; the local-time distributions are also explicit. Second, the bivariate process $(Z_t,W'_t)$ with an independent planar Brownian motion $W'$ is Harris recurrent and reversible with invariant measure $\\mu_0^{\\beta\\downarrow}\\otimes\\text{Leb}$. Third, under $P^{\\beta_i\\downarrow,i}_{z_0}$ the motion obeys the singular-drift SDE\n$$Z^j_t=z^j_0-\\frac{(1_{j=i'}-1_{j=i})}{\\sqrt{2}}\\int_0^t \\frac{\\widehat K_1(\\sqrt{2\\beta_i}|Z^i_s|)}{K_0(\\sqrt{2\\beta_i}|Z^i_s|)}\\frac{1}{Z^i_s}\\,ds+W^j_t,\\qquad 1\\le j\\le N,$$\nwhere $\\widehat K_\\nu(x)=x^\\nu K_\\nu(x)$ and $K_0,K_1$ are Macdonald functions; equivalently the interacting relative coordinate satisfies the same equation with a single singular drift. The SDE is shown to hold through and beyond the first collision time, not merely before it.","pith_inferences":["If this construction is correct, a natural candidate for the full many-$\\delta$ stochastic motion is a diffusion whose SDE carries one singular drift term per interacting pair, with the collision times handled by the same negative-moment technology; this is the author's stated plan in the companion papers, not a result proven here.","The explicit invariant density proportional to $K_0(\\sqrt{2\\beta}|z|)^2$ suggests that, under equilibrium, the interacting pair spends a substantial amount of time at small separations, consistent with an attractive contact interaction; the paper does not offer this physical interpretation.","The proof scheme—defining the law from nonzero initial data by conditioning on the pre-collision path and then using sharp negative moments to extend the SDE through the zero set—could plausibly apply to other strongly singular Schr\\\"odinger operators whose eigenfunctions have logarithmic behavior, but that extension is not attempted in this paper.","A concrete check of the claimed SDE would be to simulate the radial SDE (4.15) and compare the distribution of the inverse local time at $0$ with the gamma-subordinator identity (2.1); such a numerical test is not part of the paper."],"forward_implications":["The explicit invariant measure $\\mu_0^{\\beta\\downarrow}(dz)=\\frac{2\\beta}{\\pi}K_0(\\sqrt{2\\beta}|z|)^2\\,dz$ gives a concrete equilibrium occupation law for the interacting pair, so long-run spatial statistics of the one-delta motion are computable in closed form.","Harris recurrence of $(Z_t,W'_t)$ with invariant measure $\\mu_0^{\\beta\\downarrow}\\otimes\\text{Leb}$ implies that every set of positive invariant measure is visited infinitely often almost surely, a property directly used as the integrated recurrence bound (3.3) in the later papers of the series.","The SDE (4.1) provides a pathwise model of a single contact interaction that remains meaningful at and beyond the first collision time, because the sharp negative moments render the singular integral absolutely convergent.","The conditional laws (2.7) and (2.8) express the process before and at the collision time through ordinary Wiener measure and local time, furnishing the Feynman–Kac-type formulas that the series aims to prove for the full $N$-body delta-Bose gas."],"supporting_citations":[{"why":"Supplies the stochastic relative-motion diffusion $P^{\\beta\\downarrow}$, its skew-product representation, and the SDE for $\\mathrm{BES}(0,\\beta\\downarrow)$ that the one-delta construction builds on.","marker":"[11]"},{"why":"Constructs the special radial diffusion $\\mathrm{BES}(0,\\beta\\downarrow)$ whose inverse local time at zero is a gamma subordinator, fixing the local-time normalization used throughout.","marker":"[18]"},{"why":"Provides Erickson's continuous-extension theorem used to define the law of the diffusion at the origin by conditioning on the path up to the first hit of zero.","marker":"[19]"},{"why":"Gives resolvent solutions and semigroups for the two-dimensional two-body delta interaction, whose kernels enter the explicit transition densities (1.5) and (2.4).","marker":"[2]"},{"why":"Supplies the Laplace-inversion proof of the semigroup kernel formula and analytic regularity bounds used to prove continuity of the transition densities and the sharp negative moments.","marker":"[10]"},{"why":"Provides the integral representations and asymptotic expansions of the Macdonald functions $K_0$ and $K_1$ used in every explicit formula and moment estimate.","marker":"[29]"},{"why":"Gives the Harris-recurrence criterion for regular Feller processes invoked to prove recurrence of $(Z_t,W'_t)$ in Theorem 3.1.","marker":"[24]"},{"why":"Supplies standard facts about skew-product decompositions, local times, additive functionals, and stochastic integration used repeatedly in the proofs.","marker":"[34]"}],"fun_headline_variants":["Explicit laws for one-delta Bose motions","Singular SDE and explicit measures for delta-Bose gas","One-delta Bose: explicit density, invariant measure, SDE","Stochastic one-delta motions: explicit and singular","Explicit transition density for one-delta Bose gas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction imports the two-dimensional relative-motion diffusion from the companion paper [11], in particular the recovery of the law at the origin by conditioning on the path up to the first hit of zero; if that underlying diffusion or the conditioning does not pick out a unique process, the explicit formulas and the SDE inherit the failure.","fun_headline_variants_meta":{"raw":{"variants":["Explicit laws for one-delta Bose motions","Singular SDE and explicit measures for delta-Bose gas","One-delta Bose: explicit density, invariant measure, SDE","Stochastic one-delta motions: explicit and singular","Explicit transition density for one-delta Bose gas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1556,"prompt_tokens":1093,"completion_tokens":463,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":383}},"tokens_in":709,"tokens_out":463,"duration_ms":4601,"temperature":1.0,"reasoning_tokens":383,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:11:50.815312+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the local time at zero of the SDE (4.15) started at the origin and compare its Laplace transform with the paper's identity $\\mathbb{E}^{\\beta\\downarrow}_0[\\int_0^\\infty e^{-q\\tau}\\,dL_\\tau]=1/\\log(1+q/\\beta)$ for multiple $q$; equivalently, simulate the SDE and check its one-time marginals against the closed-form density (2.5). A mismatch at any fixed time would directly contradict Theorem 2.1 and the SDE characterization of Proposition 4.2.","supporting_citations":[{"cited_title":"and Dol´eans-Dade, C.A","cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic relative-motion diffusion $P^{\\beta\\downarrow}$, its skew-product representation, and the SDE for $\\mathrm{BES}(0,\\beta\\downarrow)$ that the one-delta construction builds on."},{"cited_title":"C4.36(β,t )","cited_arxiv_id":null,"evidence_quote":"Gives resolvent solutions and semigroups for the two-dimensional two-body delta interaction, whose kernels enter the explicit transition densities (1.5) and (2.4)."},{"cited_title":"and Watanabe, S","cited_arxiv_id":null,"evidence_quote":"Provides the integral representations and asymptotic expansions of the Macdonald functions $K_0$ and $K_1$ used in every explicit formula and moment estimate."},{"cited_title":"and Watanabe, S","cited_arxiv_id":null,"evidence_quote":"Supplies standard facts about skew-product decompositions, local times, additive functionals, and stochastic integration used repeatedly in the proofs."}],"review_version":1}