{"id":"515f41ed-9e3a-4068-aeb4-6cea8a27ee10","arxiv_id":"2607.07649","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":4,"one_line_summary":"A Feller diffusion in a finite interval with state-dependent diffusivity exhibits asymmetric escape, with inhomogeneous noise biasing trajectories toward the high-diffusivity outbreak boundary even from symmetric initial conditions.","lead":"This paper solves exactly the escape problem for a Feller diffusion (noise that vanishes near the origin) confined between two absorbing boundaries, finding that the frozen-noise region biases escape toward the far boundary. A smart generalist might read it to understand how spatially varying noise breaks symmetry in stochastic decision-making, population dynamics, and chemical kinetics.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Splitting probability (Eq. 16) is quantitatively sensitive to x_L for θ/D₀≥1 because the scale function integral diverges near the origin; qualitative asymmetry is robust but quantitative claims need a robustness check.","rationale":"The reader identified the correct load-bearing concern: the boundary placement x_L = 10⁻⁴ is not systematically checked for robustness, and it is load-bearing for the quantitative splitting probability results. I agree with this assessment. The concern is sharpest for θ/D₀ ≥ 1, where the scale function integral diverges near the origin, making the splitting probability explicitly dependent on x_L. For θ/D₀ < 1, the integral converges and results are robust. The paper's central qualitative claim—inhomogeneous diffusivity produces asymmetric escape favoring the outbreak boundary—is correct and well-supported by the exact formulas (Eqs. 12, 16) and Langevin simulations. The derivation follows standard methods (backward FPE, Laplace transform, Kummer equation) and the formulas are validated by numerical simulations with good agreement across multiple observables. The concern is quantitative, not qualitative: the specific numerical values of splitting probabilities and the exact location of the phase boundary in Fig. 7 depend on x_L for θ/D₀ ≥ 1. This does not rise to the level of requiring a verdict change. The paper makes a solid, novel contribution: exact closed-form MFPT and splitting probability for a Feller process in a finite interval, a problem not previously addressed. The formulas are fully specified and reproducible. A robustness check on x_L would strengthen the paper but is not essential for the central claims to hold. I recommend UNCHANGED: the reader's ACCEPT verdict with HIGH confidence is appropriate, with the caveat that the quantitative splitting probability results for θ/D₀ ≥ 1 should be interpreted with awareness of the x_L dependence.","tokens_in":15684,"tokens_out":9262,"duration_ms":485315,"concrete_test":"Recompute Eq. 16 for θ/D₀ = 1.0 (θ=1, D₀=1, x_R=2) with x_L ∈ {10⁻³, 10⁻⁴, 10⁻⁵, 10⁻⁶} and tabulate ε_{x_L} at x₀ = θ (symmetric start). If ε_{x_L} changes by more than 10% across this range, the quantitative splitting probability claims for θ/D₀ ≥ 1 require a caveat about x_L dependence. Also recompute the crossing point x₀* where ε_{x_L} = ε_{x_R} = 0.5 for each x_L; if x₀* shifts by more than 5% of 2θ, the phase boundary in Fig. 7 is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies x_L = 10⁻⁴ as a load-bearing choice, but the mechanism is more specific than 'ad hoc boundary.' The splitting probability (Eq. 16) is a ratio of integrals of the scale function ϕ(x) = e^{x/D₀} x^{-θ/D₀}. Near x=0, ϕ(x) ~ x^{-θ/D₀}. For θ/D₀ < 1, ∫₀ ϕ(x)dx converges, so the splitting probability has a well-defined limit as x_L→0 and results are robust. For θ/D₀ = 1, ϕ(x) ~ 1/x and the integral diverges logarithmically: the denominator in Eq. 15 scales as ln(x_R/x_L), so changing x_L from 10⁻⁴ to 10⁻⁶ shifts the denominator by ~40%. For θ/D₀ > 1 (e.g., 1.25), the integral diverges as x_L^{1-θ/D₀}, and changing x_L by two orders of magnitude changes the denominator by a factor of ~3. In both regimes, ε_{x_L}→0 as x_L→0, so the qualitative result (outbreak favored) persists, but the quantitative splitting probabilities and the crossing point where ε_{x_L}=ε_{x_R} (Fig. 6, Fig. 7 phase boundary) shift with x_L. The paper presents results for θ/D₀ = 0.5, 1.0, and 1.25 without checking this sensitivity. The MFPT (Eq. 12) is less affected because it depends on the full Green's function, not just the scale function integral, but the splitting probability is directly impacted. This does not invalidate the central qualitative claim of asymmetric escape, which is a genuine physical effect of state-dependent diffusivity and is supported by the exact formulas and simulation agreement. However, the quantitative precision of the splitting probability results for θ/D₀ ≥ 1 is contingent on the specific x_L choice.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript studies the first-passage properties of a Feller diffusion (linear drift, linear state-dependent diffusivity D(x)=D_0 x) confined to a finite interval [x_L, x_R] with two absorbing boundaries. The backward Fokker-Planck equation is solved via Laplace transform and reduction to Kummer's equation, yielding exact expressions for the mean first-passage time (MFPT, Eq. 12) and the splitting probability (Eq. 16). The central physical claim is that spatially inhomogeneous diffusivity produces asymmetric escape: the MFPT is non-monotonic with a maximum shifted toward the low-noise (extinction) boundary, and the splitting probability favors the outbreak boundary even from a symmetric initial position. The analytical results are validated against Langevin simulations (2-5 x 10^5 trajectories, dt=10^-4). The paper also presents phase diagrams for the coefficient of variation and the splitting probability difference, and discusses effective outcome rates in a speed-accuracy trade-off context.","tokens_in":16044,"tokens_out":1318,"duration_ms":290723,"significance":"The paper provides exact, closed-form expressions for the MFPT and splitting probabilities of a bounded Feller process, a problem that, to my knowledge, has not been previously solved in this two-boundary setting. The derivation is self-contained and parameter-free in the sense that theta and D_0 are treated as control variables, not fitted to data. The physical effect—inhomogeneous diffusivity biasing escape toward the high-noise boundary—is a genuine and clearly demonstrated phenomenon. The comparison with the Ornstein-Uhlenbeck process is a useful framing device. The simulation data agree well with the analytical curves, supporting the correctness of the derivation.","major_comments":[{"comment":"§V, Eq. (16): The splitting probability is expressed as a ratio of integrals of the scale function phi(x) = e^{x/D_0} x^{-theta/D_0}. Near x=0, phi(x) ~ x^{-theta/D_0}. For theta/D_0 >= 1, the integral of phi(x) from x_L to any finite x diverges as x_L -> 0. Concretely, for theta/D_0 = 1, the denominator of Eq. (16) scales as ln(x_R/x_L), so changing x_L from 10^{-4} to 10^{-6} shifts the denominator by ~40%. For theta/D_0 = 1.25, the integral diverges as x_L^{1-theta/D_0}, and a two-order-of-magnitude change in x_L changes the denominator by a factor of ~3. The paper presents results for theta/D_0 = 0.5, 1.0, and 1.25 (Figs. 4, 6, 7) without checking this sensitivity. The qualitative asymmetry (outbreak favored) is robust because epsilon_{x_L} -> 0 as x_L -> 0 in all regimes, but the quantitative splitting probabilities and the phase boundary in Fig. 7 (where Delta_epsilon = 0) shift. A","section":null},{"comment":"robustness check—repeating the key splitting-probability results (Fig. 6 for theta/D_0 = 1.0 and 1.25, and the phase boundary in Fig. 7) for at least one additional value of x_L (e.g., 10^{-2} and 10^{-6})—is needed to establish the quantitative claims. This does not affect the MFPT (Eq. 12) as strongly, since it depends on the full Green's function rather than just the scale-function integral, but the splitting probability is directly impacted. The central qualitative claim of asymmetric escape is not invalidated; the concern is about the precision and parameter-dependence of the quantitative results.","section":null}],"minor_comments":[{"comment":"§III: The choice x_L = 10^{-4} is described as 'x_L ~ 0.' For theta/D_0 >= 1, the origin is an entrance boundary (inaccessible), so placing an absorbing boundary at a small but finite x_L is physically distinct from placing it at the origin. This distinction should be stated explicitly.","section":null},{"comment":"Fig. 3: The x-axis label 'x_L/2theta' is confusing; it appears to be a scaled initial position x_0/(2theta) but the label suggests it is a ratio involving x_L. Please clarify or correct the axis label.","section":null},{"comment":"§VI, last paragraph: 'initialized in a position inclined to the hell (extinction) site' — 'hell' appears to be a typo for 'well' or should be rephrased.","section":null},{"comment":"Eq. (16): The upper incomplete gamma function Gamma(a, x) is defined with integration from x to infinity, but the arguments involve negative x values (e.g., -x_R/D_0). The use of incomplete gamma functions with negative arguments should be clarified, perhaps noting the analytic continuation or the relation to the exponential integral.","section":null},{"comment":"Appendix B, Eq. (B3): The denominator '2(H'_L - H'_R) + M'_L - M'_R' lacks parentheses or clear grouping. Please verify the formula and add parentheses for clarity.","section":null},{"comment":"§II, Eq. (2): The noise term is written as sqrt(2D(x(t))) xi(t), but the FPE (Eq. 3) has D_0 as the coefficient. The relation D(x) = D_0 x is stated, but the factor of 2 in the Langevin equation versus the FPE should be checked for consistency (Itô convention).","section":null}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about x_L sensitivity is valid and specific to the splitting probability for theta/D_0 >= 1. The MFPT is less affected. The authors should be able to address this with a straightforward numerical robustness check. The paper is otherwise a solid contribution with exact results in a previously unexplored setting."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and for identifying a genuine sensitivity in our splitting-probability results. The referee's mathematical analysis of the scale-function integral is correct, and we will incorporate the requested robustness checks in the revised manuscript.","responses":[{"response":"The referee's analysis is mathematically correct, and we acknowledge this gap in our presentation. The scale function φ(x) = e^{x/D₀} x^{-θ/D₀} indeed has a non-integrable singularity at x = 0 when θ/D₀ ≥ 1, so the denominator of Eq. (16) is sensitive to the precise value of x_L in this regime. For θ/D₀ = 1, the logarithmic scaling ln(x_R/x_L) means that a two-order-of-magnitude change in x_L produces a ~40% shift in the denominator, and for θ/D₀ = 1.25 the power-law divergence x_L^{1-θ/D₀} produces an even larger shift (~factor of 3). We agree that this sensitivity should have been explicitly addressed. We will add a robustness check in the revised manuscript: we will repeat the splitting-probability results of Fig. 6 (for θ/D₀ = 1.0 and 1.25) and the phase boundary of Fig. 7 for at least two additional values of x_L (e.g., 10^{-2} and 10^{-6}), and we will add a discussion of the x_L-dependence of the quantitative splitting probabilities. We note that the referee is also correct that the qualitative asymmetry—outbreak favored from symmetric initial conditions—is robust, since ε_{x_L} → 0 as x_L → 0 in all regimes. The central physical claim of the paper is not affected; what needs to be made explicit is the parameter-dependence of the quantitative results.","revision_made":"yes","referee_comment":"§V, Eq. (16): The splitting probability is expressed as a ratio of integrals of the scale function phi(x) = e^{x/D_0} x^{-theta/D_0}. Near x=0, phi(x) ~ x^{-theta/D_0}. For theta/D_0 >= 1, the integral of phi(x) from x_L to any finite x diverges as x_L -> 0. [...] The paper presents results for theta/D_0 = 0.5, 1.0, and 1.25 (Figs. 4, 6, 7) without checking this sensitivity."},{"response":"We agree entirely and will perform the requested robustness check. Specifically, we will: (i) recompute the splitting probabilities of Fig. 6 for θ/D₀ = 1.0 and 1.25 at x_L = 10^{-2}, 10^{-4}, and 10^{-6}, showing the quantitative shifts; (ii) recompute the Δε = 0 phase boundary of Fig. 7 for the same additional x_L values; and (iii) add a paragraph in §V discussing the x_L-sensitivity, including the asymptotic behavior of the denominator integral in the different θ/D₀ regimes. We also confirm the referee's observation that the MFPT (Eq. 12) is less directly affected, since it depends on the full Green's function rather than solely on the scale-function integral. We will add a brief remark to that effect as well.","revision_made":"yes","referee_comment":"A robustness check—repeating the key splitting-probability results (Fig. 6 for theta/D_0 = 1.0 and 1.25, and the phase boundary in Fig. 7) for at least one additional value of x_L (e.g., 10^{-2} and 10^{-6})—is needed to establish the quantitative claims. This does not affect the MFPT (Eq. 12) as strongly..."}],"tokens_in":15474,"tokens_out":1220,"duration_ms":98006,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Short version: this paper solves the two-boundary first-passage problem for a Feller diffusion, getting exact closed-form MFPT (Eq. 12) and splitting probability (Eq. 16) in terms of Kummer functions and incomplete gamma functions. That is genuinely new — prior work by Masoliver and Ray handled only single-boundary escape — and the derivation is clean and correct. The physical picture is simple and well-argued: state-dependent diffusivity D(x) = D₀x vanishes near the origin, creating a frozen bottleneck that biases escape toward the high-noise (outbreak) boundary even from a symmetric starting point. Langevin simulations with 2–5×10⁵ trajectories confirm the analytics quantitatively across three regimes of θ/D₀. The CV analysis and the phase diagrams (Figs. 5, 7) are a nice touch and give the reader useful intuition for where fluctuations dominate the mean. The comparison to the OU process throughout is the right baseline and makes the asymmetry concrete. The speed-accuracy / decision-making framing at the end is somewhat speculative but does not detract from the core results. The one real soft spot is the boundary placement x_L = 10⁻⁴. The stress-test note is right that this matters quantitatively for the splitting probability when θ/D₀ ≥ 1: the scale function ϕ(x) ~ x^{-θ/D₀} near the origin, so the denominator in Eq. 15 is sensitive to x_L in that regime. For θ/D₀ = 1 it diverges logarithmically; for θ/D₀ = 1.25 it diverges as a power law. Changing x_L by two orders of magnitude can shift the splitting probability by tens of percent. The qualitative asymmetry (outbreak favored) is robust because ε_{x_L} → 0 as x_L → 0 regardless, but the quantitative values in Figs. 6–7 and the phase boundary location depend on the specific x_L choice. The paper does not check this. The MFPT (Eq. 12) is less affected because it depends on the full Green's function, not just the scale function integral. This is a fixable issue — a short paragraph or a figure showing splitting probability vs. x_L for the three θ/D₀ values would close it. It does not undermine the central claim. This paper is for people working in first-passage theory and stochastic processes with multiplicative noise. It deserves a serious referee who should ask for the x_L robustness check but otherwise should not hold it up.","headline":"Exact MFPT and splitting probabilities for a Feller process in a bounded interval — a clean, novel result with one quantitative soft spot near the singular boundary.","tokens_in":16807,"tokens_out":602,"would_cite":true,"duration_ms":99063,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Frozen noise near one boundary biases escape toward the other","keywords":[],"falsifier":"If the extinction boundary x_L is moved to a position where the diffusion coefficient is not near-zero (e.g., x_L = 0.1 or 0.5), the asymmetry in splitting probability should weaken or disappear, with the splitting probabilities approaching the symmetric Ornstein-Uhlenbeck result.","tokens_in":15755,"feed_emoji":"🎲","tokens_out":1118,"duration_ms":159996,"temperature":0.7,"pith_summary":"This paper studies a Feller diffusion process confined to a finite interval with two absorbing boundaries placed at equal potential-energy distances from a stable point. The key feature is that the diffusion coefficient is proportional to position, vanishing near the origin. This creates spatially inhomogeneous noise: fluctuations freeze near the left (extinction) boundary and grow toward the right (outbreak) boundary. The authors derive exact expressions for the mean first-passage time (MFPT) and splitting probability using Kummer and Tricomi confluent hypergeometric functions, and confirm them with Langevin simulations. The central result is that this inhomogeneous noise breaks the symmetry one would expect from the potential landscape alone. Even when the process starts from a position equidistant from both boundaries, the splitting probability favors the outbreak boundary because the frozen noise near the extinction boundary acts as a kinetic bottleneck, reflecting trajectories back toward the high-noise region. The MFPT is non-monotonic in the initial condition, with its maximum shifted toward the low-noise (extinction) side rather than sitting at the potential minimum. The coefficient of variation of escape times reaches its minimum at this same shifted location. The authors also map these results onto a speed-accuracy trade-off framework, defining effective extinction and outbreak rates as ratios of splitting probability to mean escape time, and showing that the outbreak rate dominates across most initial conditions.","feed_headline":"State-dependent noise breaks escape symmetry in Feller diffusion","feed_subtitle":"When diffusivity vanishes near one boundary, a Feller process favors the other exit even from a symmetric start, shifting the longest escape","key_machinery":"Backward Fokker-Planck equation reduced to Kummer's equation; exact MFPT in terms of derivatives of confluent hypergeometric functions M and U; splitting probability in terms of incomplete gamma functions; comparison with Ornstein-Uhlenbeck process (homogeneous noise) as a symmetric baseline.","core_discovery":"The central object is the Feller diffusion process, a one-dimensional Markov process with linear drift toward a stable point and a diffusion coefficient proportional to position that vanishes at the origin. When confined between two absorbing boundaries at equal potential-energy distance from the stable point, the state-dependent diffusivity produces asymmetric escape: the splitting probability favors the high-noise (outbreak) boundary even from a symmetric starting position, and the MFPT maximum shifts toward the low-noise (extinction) boundary. The exact MFPT is expressed in closed form via derivatives of Kummer and Tricomi confluent hypergeometric functions, and the splitting probability,","pith_inferences":["The choice of x_L = 10^{-4} as the extinction boundary places it very close to the singular point x=0 where the diffusion coefficient vanishes. If x_L were moved further from the origin, the asymmetry in splitting probability would likely weaken, because the noise would not be as severely frozen near the boundary. A systematic study of how escape statistics vary with x_L would clarify whether the ","The authors note that for theta/D_0 > 1, the origin becomes an inaccessible entrance boundary. This suggests a sharp transition in escape behavior: for theta/D_0 crossing unity, the nature of the extinction boundary changes qualitatively, which could produce non-analytic behavior in the splitting probability or MFPT as a function of this ratio.","The connection to speed-accuracy trade-offs in decision-making is drawn by analogy. A more direct test would be to embed this Feller process into a sequential-sampling framework and compare predicted decision-time distributions and error rates against experimental data from perceptual decision tasks."],"forward_implications":["In population dynamics models with density-dependent demographic noise, extinction may be less likely than outbreak even when the deterministic force field is symmetric, because noise suppression at low population sizes creates a kinetic bottleneck.","For decision-making models that use diffusion processes with two absorbing boundaries, spatially varying noise can bias decisions toward the high-noise boundary, suggesting that speed-accuracy trade-offs are sensitive to the noise geometry, not just the drift.","The phase diagram for the coefficient of variation (Fig. 5) identifies parameter regimes where escape-time fluctuations dominate the mean, which could inform when noise-control interventions such as stochastic resetting would be most effective.","The result that the MFPT maximum and CV minimum are shifted from the potential minimum toward the low-noise boundary provides a diagnostic signature: observing this shift in experimental escape-time data would indicate state-dependent rather than homogeneous noise."],"fun_headline_variants":["Noise that fades at one boundary steers Feller escape toward the other exit","Asymmetric escape in Feller diffusion driven by state-dependent diffusivity","Where diffusivity vanishes, Feller escape tilts toward the noisy boundary","Position-dependent noise biases Feller diffusion toward outbreak exit","Feller escape maximum shifts toward low-noise boundary under multiplicative noise"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The extinction boundary is placed at x_L = 10^{-4}, very close to the singular point x=0 where the diffusion coefficient vanishes. All quantitative results depend on this specific choice, and the paper does not systematically verify whether the asymmetries persist if the boundary is moved further from or closer to the singular point.","fun_headline_variants_meta":{"raw":{"variants":["Noise that fades at one boundary steers Feller escape toward the other exit","Asymmetric escape in Feller diffusion driven by state-dependent diffusivity","Where diffusivity vanishes, Feller escape tilts toward the noisy boundary","Position-dependent noise biases Feller diffusion toward outbreak exit","Feller escape maximum shifts toward low-noise boundary under multiplicative noise","Closed-form exit times reveal noise-driven asymmetry in Feller diffusion","State-dependent noise favors outbreak exit even from symmetric Feller start","Multiplicative noise breaks escape symmetry in confined Feller diffusion","Fading noise at extinction boundary redirects Feller escape to outbreak state","Feller diffusion escape asymmetry tied to vanishing diffusivity at one boundary"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1010,"prompt_tokens":540,"completion_tokens":470,"prompt_tokens_details":null},"tokens_in":540,"tokens_out":470,"duration_ms":20651,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T03:37:48.023901+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If the extinction boundary x_L is moved to a position where the diffusion coefficient is not near-zero (e.g., x_L = 0.1 or 0.5), the asymmetry in splitting probability should weaken or disappear, with the splitting probabilities approaching the symmetric Ornstein-Uhlenbeck result.","supporting_citations":[],"review_version":1}