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REVIEW 2 major objections 6 minor 91 references

Feller diffusion in an interval: Inhomogeneous fluctuation-induced asymmetric escape

T0 review · 2 major / 6 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Frozen noise near one boundary biases escape toward the other

desk verdict 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. read the letter →

arxiv 2607.07649 v1 pith:BO5S2L4W submitted 2026-07-08 cond-mat.stat-mech

Prashanta Bauri , Debasish Mondal This is my paper · ORCID
classification cond-mat.stat-mech
keywords escapeexitextinctioninitialintervaloutbreakpotentialtime
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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,

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

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.

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 (2)
  1. §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
  2. 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.
minor comments (6)
  1. §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.
  2. 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.
  3. §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.
  4. 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.
  5. 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.
  6. §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).

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: §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.

    Authors: 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: yes

  2. Referee: 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...

    Authors: 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: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: MFPT and splitting probability derived from standard Feller SDE without fitted parameters or self-citation chains

full rationale

The derivation is self-contained. The starting point (Eq. 1) is the standard Feller process SDE from the literature (Feller 1951, Capocelli & Ricciardi 1973), not an ansatz defined in terms of the target results. The MFPT (Eq. 12) is derived from the backward Fokker-Planck equation (Eq. 7) via Laplace transform, yielding a Kummer equation whose solution with absorbing boundary conditions gives the closed-form result. The splitting probability (Eq. 16) is derived from the standard scale-function integral (Eq. 15), a textbook method (Gardiner, Redner), yielding incomplete gamma functions. No parameters are fitted to data and then re-presented as predictions: θ and D₀ are treated as control variables, and all analytical results are confirmed against independent Langevin simulations (Figs. 2-8). The comparison to the OU process uses independently known results (Appendix B). Self-citations (refs 37, 39, 40, 79, 81, 82, 83) appear in the introduction and conclusion for context on prior work by the authors on related topics (noise-induced transitions, stochastic resetting), but none are load-bearing for the central derivation. The boundary choice x_L = 10⁻⁴ is a numerical parameter choice, not a fitted input renamed as a prediction. The derivation chain from SDE → backward FPE → Laplace transform → Kummer equation → MFPT, and SDE → scale function → splitting probability, contains no step that reduces to its own inputs by construction.

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

The paper introduces no new physical entities, particles, forces, or dimensions. The Feller process, the harmonic potential, and the absorbing boundaries are all standard constructs. The 'extinction' and 'outbreak' labels are interpretive framings of the two absorbing boundaries, not new entities. The Kummer and Tricomi confluent hypergeometric functions and incomplete gamma functions are standard mathematical objects.

free parameters (4)
  • θ (drift parameter / potential minimum location) = varied (e.g., θ=1 in Figs. 3, 6, 8; θ/D₀ = 0.5, 1.0, 1.25 in Fig. 4)
    Control parameter determining the depth and location of the potential well; not fitted to data but varied to explore regimes.
  • D₀ (noise amplitude parameter) = 1.0 (default)
    Controls the strength of multiplicative fluctuations; set to 1.0 for most results, not fitted.
  • x_L (left absorbing boundary position) = 10^{-4}
    Chosen to be near the singular point x=0; not systematically varied. This is a modeling choice rather than a fitted parameter.
  • x_R (right absorbing boundary position) = 2θ
    Set to twice the potential minimum to create approximately symmetric potential energy at both boundaries.
assumptions (4)
  • domain assumption The Feller SDE (Eq. 1) in Itô interpretation correctly models the physical process.
    The Itô interpretation is standard for the Feller process; the Stratonovich interpretation would yield different noise-induced drift terms. Stated in Section II.
  • ad hoc to paper The boundary at x_L = 10^{-4} is a valid approximation of a boundary near the singular point x=0.
    Section III states x_L ~ 0 (10^{-4}) without systematic robustness checks. The Feller process has a degenerate diffusion coefficient at x=0, and the boundary classification depends on θ/D₀.
  • domain assumption The improved Euler method with dt=10^{-4} accurately captures the dynamics near the singular point.
    Section III states this choice; the agreement with analytical results in Figs. 2-4, 6, 8 is used as justification, but no convergence study is shown.
  • domain assumption The effective outbreak/extinction rates (ER=ε_xL/⟨τ⟩, OR=ε_xR/⟨τ⟩) are meaningful analogs of speed-accuracy trade-off ratios.
    Section V defines these rates and draws the analogy to decision-making theory; the mapping is qualitative and not derived from a decision-making model.

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Pith. "Pith review of Feller diffusion in an interval: Inhomogeneous fluctuation-induced asymmetric escape." pith.science (2026). https://pith.science/paper/BO5S2L4W

@misc{pith2026260707649,
  author       = {Pith},
  title        = {Pith review of: Feller diffusion in an interval: Inhomogeneous fluctuation-induced asymmetric escape},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BO5S2L4W}},
  note         = {Machine review of arXiv:2607.07649}
}
read the original abstract

We present an inhomogeneous fluctuation-induced asymmetric escape event of a Feller diffusion confined to a finite interval with two competing absorbing boundaries. The dynamics correspond to an overdamped Brownian motion in a shifted harmonic potential with state-dependent diffusivity. We set two exit points (equal potential energies), equidistant from the potential minima: an extinction site near the origin and an outbreak point. The multiplicative fluctuations are suppressed near the extinction boundary, biasing trajectories toward the outbreak state. The mean exit time exhibits a non-monotonic dependence on the initial condition and the drift-to-noise strength ratio, attaining a maximum when the particle is initialized toward the extinction (low-noise) site. The outbreak possibility is more likely even if the process started with a small initial bias towards the low-noise boundary. The spatial location of the lowest coefficient of variation (CV) is nicely corroborated by the maximum exit time, which is the hallmark of a stochastic escape from an interval. The fluctuation in escape time is found to dominate its mean as a non-trivial function of the drift-to-noise strength and the initial spatial bias. The observed asymmetries in escape also shape the speed-accuracy trade-off in stochastic decision-making events.

Figures

Figures reproduced from arXiv: 2607.07649 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of the potential energy set-up [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Variation of the stationary probability distribution [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison plot between homogeneous (open mark [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Variation of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison plot between homogeneous (open mark [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Phase diagram of the splitting probability difference [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Effective outbreak (circular points) or extinction (tri [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

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