Pith. sign in

REVIEW 3 major objections 3 minor 32 references

Some harmonic functions for killed Markov branching processes with immigration and culling

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For killed continuous-time branching processes with immigration and culling, the paper constructs explicit bounded harmonic functions and derives the Laplace transforms of downward first-passage times and of the explosion time, whenever…

desk verdict The paper's central explicit harmonic function Φ_q in Theorem 3.6(i) has the exponential sign reversed, so as printed the defining integral diverges in allowed cases; the corrected sign is what the proofs actually use, so the paper needs a substantive typo-fix revision rather than a skip. read the letter →

arxiv 1908.04714 v5 pith:NE4JKCOB submitted 2019-08-13 math.PR

Matija Vidmar This is my paper · ORCID
classification math.PR MSC 60J8060J50
keywords continuous-timeBienaymé-Galton-WatsonprocessbranchingwithimmigrationandcullingkilledMarkovharmonicfunctionfirstpassagedownwardsexplosiontimeLaplacetransformskip-freechain
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

The paper aims to identify, in closed form, the bounded harmonic functions of a continuous-time branching process with immigration and culling after the process is killed at a constant rate $q$ and stopped at extinction or explosion. From these functions it derives the Laplace transforms of the first passage time to low levels and of the explosion time, and then probabilities and expectations by letting $q$ tend to $0$. The identification succeeds whenever the killing rate is large enough relative to the branching and culling mechanisms; this restriction always holds for non-supercritical branching or when there is no culling. The value of the work is the level of explicitness: the harmonic functions are written as one-dimensional integrals in terms of the offspring and immigration/culling probability generating functions.

What carries the argument

The machinery is an integral ansatz $f(x)=\int_\alpha^\beta w(v)v^x\,dv$ for the bounded solutions of the generator equation $(q+\lambda x+\mu)f(x)=\lambda x\sum p_kf(x+k-1)+\mu\sum r_kf(x+k)$. Plugging the ansatz in and integrating by parts reduces the equation to the first-order ODE $(q+\mu(1-\tilde r(v)))w(v)=-\frac{d}{dv}[\lambda(\tilde p(v)-v)w(v)]$, whose solution is expressed through $\rho(v)=\lambda|\tilde p(v)-v|$ and $\gamma_q(v)=(q+\mu(1-\tilde r(v)))/\rho(v)$. The choice of integration limits, $0$ to $\varphi$ for $\Phi_q$ and $\varphi$ to $1$ for $\Psi_q$, makes the boundary terms vanish and selects the harmonic function with the required behaviour at infinity. The downward skip-free property of the process is what lets the single-level exit probabilities be recovered from the ratio $f(x)/f(a)$.

What would settle it

Take a concrete supercritical binary branching mechanism with $p_0=1/3$, $p_2=2/3$, $\lambda=3$, and culling with $r_{-1}=1$, $\mu=1$; here $\varphi=1/2$ and $\varphi_q=1/(q+1)$. Choose $q=2$, so $\varphi_q=1/3<\varphi$. Solve the linear system (3.2) on a truncated state space for the bounded solution $f$ with $f(\infty)=0$, compute the ratio $f(1)/f(0)$, and compare it with $\Phi_q(1)/\Phi_q(0)$ from Theorem 3.6(i). Any discrepancy beyond numerical error would refute the formula.

Watch

Extended reading notes

Core claim

The central discovery is a pair of explicit harmonic functions for the killed process $X^q$. Under the condition $\varphi_q<\varphi$, where $\varphi_q$ is the root of $q=\mu(\tilde r(z)-1)$ in $(0,1)$ and $\varphi$ is the extinction root of $\tilde p(z)=z$, the function $$\Phi_q(x)=q\$int_0^{{\varphi}}$ \frac{\exp\{-\$int_v^{{\varphi_q}}$\gamma_q(w)\,dw\}}{\rho(v)} v^x\,dv$$ belongs to the one-dimensional space of bounded harmonic functions that vanish at infinity. Under the explosivity condition, a second function $$\Psi_q(x)=1-q\int_{\varphi}^{1} \frac{\exp\{-\$int_v^{1}$\gamma_q(w)\,dw\}}{\rho(v)} v^x\,dv$$ spans the complementary direction. Together with the martingale property of these functions, this yields $P_x[e^{-qT_a^-};T_a^-<\infty]=\Phi_q(x)/\Phi_q(a)$ and, under explosivity, $P_x[e^{-q\zeta};\zeta<T_a^-]=\Psi_q(x)-\Psi_q(a)\Phi_q(x)/\Phi_q(a)$. The formulas are explicit in the probability generating functions of the branching and immigration/culling mechanisms, and they persist at $q=0$ whenever $\varphi\le\varphi$.

Load-bearing premise

Everything hinges on the strict inequality $\varphi_q<\varphi$ (equivalently, when there is culling and supercritical branching, on $q>\mu(\tilde r(\varphi)-1)$), which makes the exponential weight in the integral for $\Phi_q$ decay fast enough at $\varphi$ and forces $\Phi_q(\infty)=0$; if that fails, no explicit bounded harmonic function is produced and the Laplace-transform formulas are not established.

Editorial extensions

If this is right

  • The Laplace transform of every downward first-passage time is given in closed form for all killing rates $q$ with $\varphi_q\le\varphi$, including $q=0$ whenever $\varphi\le\varphi$, so extinction probabilities become explicitly computable from the two probability generating functions.
  • When the process can explode, the Laplace transform of the explosion time before hitting level $a$ is also explicit, and its $q\downarrow 0$ limit shows that for $\varphi\le\varphi$ explosion and extinction before passage downwards exhaust the probability space.
  • Means of first-passage times and of the joint lifetime/infimum quantities follow by differentiating the transforms, as in the paper's Corollaries 5.2 and 5.3.
  • A Doob transform by $\Phi_q$ gives the law of the process conditioned to hit $0$ before an independent exponential clock rings, with explicit jump rates.
  • The excursion decomposition at the minimum yields the joint law of the last minimum before the exponential time, the time to reach it, and the remaining time, with explicit formulas involving $\Phi_q$.

Reading between the lines

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

  • The same integral-and-ODE route suggests that a full family of scale functions for these processes could be built from the same $\rho$ and $\gamma_q$, which would open the two-sided exit problem; the paper only identifies the minimal scale function corresponding to downward passage.
  • Because the formulas hold on a neighbourhood of infinity in $q$, a Laplace-transform uniqueness argument makes the laws of $T_a^-$ and $\zeta$ determined; one could therefore invert the transforms numerically to obtain density approximations for concrete parameter choices.
  • For supercritical branching with culling and small $q$, when $\varphi_q\ge\varphi$, the paper produces no explicit bounded harmonic function; analytic continuation from the large-$q$ formulas sometimes recovers the transforms, but constructing the harmonic function itself in that regime appears to need a genuinely different idea.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies continuous-time Bienaymé-Galton-Watson processes with immigration and culling, with 0 absorbing, when additionally killed at rate q and stopped on extinction or explosion. The main object is the identification, under the condition φ_q < ϕ, of explicit harmonic functions of the killed process: a function Φ_q that vanishes at infinity and a function Ψ_q used when explosions can occur. From these harmonic functions the author derives Laplace transforms (at argument q) of the first-passage times downwards and of the explosion time, and then a number of consequences: extinction/explosion probabilities via q↓0, means of first-passage and explosion times, a Doob transform conditioning on extinction before an exponential time, a factorization at the minimum, and an extension to the cumulative-lifetime-to-date process.

Significance. If the stated formulas are correct, the paper provides genuinely explicit, parameter-free harmonic functions for a class of killed Markov branching processes with immigration and culling, and thereby the Laplace transforms of the first-passage downwards and explosion times under the stated restriction on the killing rate. This goes beyond earlier work on pure continuous-time branching processes and parallels results for continuous-state branching processes with immigration. The main structural idea—using the downwards skip-free property and an integral ansatz—is sound and is clearly explained. However, as printed, the central display in Theorem 3.6 contains a sign error that makes the displayed integral divergent in an allowed case, and the same sign error propagates to Corollary 4.14. Because the advertised explicitness of the paper rests on these displays, this must be corrected before the paper can be accepted.

major comments (3)
  1. [Theorem 3.6(i), proof of Theorem 3.6] The displayed formula for Φ_q has the sign of the inner integral reversed. For ω(v)=exp{−∫_v^{φ_q} γ_q}, the derivative is ω′(v)=+γ_q(v)ω(v), not −γ_q(v)ω(v) as the proof states. The ODE derived from (3.2) for w(v)=ρ(v)^{-1} exp{−∫_v^{φ_q} γ_q} would be (ρw)′=+γ_q ρw, whereas the correct equation is (ρw)′=−γ_q ρw. The correct weight is exp{−∫_{φ_q}^v γ_q}, equivalently exp{+∫_v^{φ_q} γ_q}. As printed, the integral defining Φ_q is not finite in allowed cases: take μ=0, p0=p2=1/2, λ=1, and q>0; then φ_q=0, the branching root is ϕ=1, and the displayed integrand is 2q(1−v)^{-2} exp{2qv/(1−v)}, which diverges at v=1. This is load-bearing because Φ_q is the harmonic function used in Theorem 4.2. The corrected formula is already implicit in Remark 3.12, but the theorem statement, the definition of ω in the proof, and the asserted ODE are internally inconsistent as written.
  2. [Theorem 3.6, Ψ_q display] The same sign error appears in the formula for Ψ_q. The expression as printed contains exp{−∫_v^1 γ_q}, whereas the integration-by-parts computation in Remark 3.12 uses exp{−∫_1^v γ_q} (equivalently exp{+∫_v^1 γ_q}). With the printed sign, the function does not satisfy the harmonic equation (3.2) on (ϕ,1); with the corrected sign, it matches the stated expression in Remark 3.12 and the limiting argument in the proof of Theorem 4.2. This is another load-bearing sign error in the main theorem.
  3. [Corollary 4.14] The display for Φ_{q,q̄} contains the same sign reversal in the inner integral, with ∫_v^{φ_q} instead of ∫_{φ_q}^v in the exponential. In the same example as above, the displayed integrand would fail to be integrable, so the formula for the joint Laplace transform in (4.6) is not established as printed. The correction should be made consistently with Theorem 3.6.
minor comments (3)
  1. [Section 3, Definition 3.5 vs. Theorem 3.6] The notation is confusing because the culling root φ and the branching extinction root ϕ look almost identical. Definition 3.5 sets φ=0 when μr_{-1}=0, while Theorem 3.6 uses ϕ in the integration limits; the two symbols are distinguished only by a subtle glyph. The authors should use visibly distinct names, for example φ^br and φ^cu or explicit verbal reminders.
  2. [Proof of Theorem 3.6] The proof says “Set ω := exp{−∫_·^{φ_q} γ_q}. Then … it solves the o.d.e. −γ_q ω = ω′.” These two statements cannot both be true; the derivative is +γ_q ω. This is not merely a typo in a peripheral sentence, because the same inconsistency is exactly what produces the wrong sign in the main display.
  3. [Remark 4.3] The analytic-continuation remark is a useful complement to the theorem, but the phrase “provided that one is able to recognize the r.h.s. of (4.1) as f(q) for an analytic function” could be made more precise, since the right-hand side as a function of q involves integrals with q-dependent endpoints and integrands. This is a clarity issue, not a mathematical objection.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the harmonic functions are verified directly against the generator equation; the sign reversal in the printed Phi_q is a correctness issue, not a circular input-output reduction.

full rationale

The derivation chain is not circular in the sense of the rubric. Theorem 3.6 constructs Phi_q and Psi_q by inserting the ansatz f(x)=integral w(v) v^x dv into the generator equation (3.2), solving the resulting first-order ODE for w, and then checking the martingale equation via Proposition 3.1(iv); the Laplace-transform conclusions of Theorem 4.2 are then read off from the general characterization in Proposition 3.3, with no parameter fitted to the target transform. The root phi_q of Lemma 3.4 is imported with the proof line 'These are well-known facts from the theory of (homogeneous-Poisson-process-subordinated) left-continuous random walks, see e.g. [4, 31]'; although one of the cited works is by the same author, the root equation is a standard external fluctuation fact and is not the result being derived, so this is at most a minor self-citation and does not make the central claim circular. I note separately, because it is in the manuscript, that the printed exponent in Theorem 3.6(i) has a sign reversal: with omega(v)=exp{-integral_v^{phi_q} gamma_q}, one has omega'=gamma_q omega, not -gamma_q omega, so the printed integrand is not the solution of the ODE used in the proof, and in the mu=0 example the integral diverges. This is a load-bearing mathematical defect and needs correction, but it is an inconsistency, not an equivalence-by-construction, and therefore does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new postulated entities. The central result rests on standard Markov chain theory, the specific model definition, and the external characterization of φ_q (Lemma 3.4). The only 'ad hoc' ingredient is the ansatz f(x)=∫w(v)v^x dv, which is a motivational device, not an unproved assumption; the resulting functions are verified directly.

assumptions (6)
  • domain assumption The process X is a minimal càdlàg continuous-time Markov chain with generator Q given by Q00=0 and for n∈N, -Qnn=λn+µ, Qn(n+m)=p_{m+1}λ + r_m µ for m∈N∪{-1}; 0 is absorbing.
    Specifies the class of processes; all results are conditional on this setup (Section 2).
  • domain assumption The state space is N0 with cemetery ∞, and the process is stopped on extinction or explosion.
    Standard construction of minimal ctMc; used throughout the paper.
  • domain assumption Standing assumptions: p0>0, λ>0, and the path is not a.s. nonincreasing (so either p0+p1<1, or µ>0 and r_{-1}<1).
    Stated in Section 1.1; ensures nontrivial branching and well-defined hitting probabilities.
  • standard math Without loss of generality p1=0, by reparameterizing.
    Section 2 states the general case reduces to this by changing the rate to λ(1-p1).
  • standard math Standard facts about continuous-time Markov chains, martingale optional stopping, and the martingale problem for minimal processes.
    Used in Proposition 3.1 and throughout; not proven in the paper.
  • standard math Lemma 3.4: the equation q=µ(ṙ(z)-1) has a unique root φ_q in (0,1), and q↦φ_q is a decreasing bijection onto (0,φ).
    Stated as well-known from the theory of homogeneous-Poisson-process-subordinated left-continuous random walks, see [4,31]. It is external to the paper and not reproven.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Some harmonic functions for killed Markov branching processes with immigration and culling." pith.science (2026). https://pith.science/paper/NE4JKCOB

@misc{pith2026190804714,
  author       = {Pith},
  title        = {Pith review of: Some harmonic functions for killed Markov branching processes with immigration and culling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NE4JKCOB}},
  note         = {Machine review of arXiv:1908.04714}
}
abstract

For a continuous-time Bienaym\'e-Galton-Watson process, $X$, with immigration and culling, $0$ as an absorbing state, call $X^q$ the process that results from killing $X$ at rate $q\in (0,\infty)$, followed by stopping it on extinction or explosion. Then an explicit identification of the relevant harmonic functions of $X^q$ allows to determine the Laplace transforms (at argument $q$) of the first passage times downwards and of the explosion time for $X$. Strictly speaking, this is accomplished only when the killing rate $q$ is sufficiently large (but always when the branching mechanism is not supercritical or if there is no culling). In particular, taking the limit $q\downarrow 0$ (whenever possible) yields the passage downwards and explosion probabilities for $X$. A number of other consequences of these results are presented.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [3]

    Avram, P

    F. Avram, P. Patie, and J. W ang. Purely excessive function s and hitting times of continuous-time branching processes . Methodology and Computing in Applied Probability , 21(2):391–399, 2019

    work page 2019

  2. [12]

    Duhalde, C

    X. Duhalde, C. Foucart, and M. Ma. On the hitting times of c ontinuous-state branching processes with immigration. Stochastic Processes and their Applications , 124(12):4182 – 4201, 2014

    work page 2014

  3. [1]

    Asmussen and H

    S. Asmussen and H. Hering. Branching Processes. Progress in probability and statistics. Birkh¨ auser, 198 3

    work page

  4. [2]

    K. B. Athreya and P. E. Ney. Branching Processes. Dover Books on Mathematics. Dover Publications, 2004

    work page 2004

  5. [4]

    Avram and M

    F. Avram and M. Vidmar. First passage problems for upwards skip-free random walks via the scale functions paradigm. Advances in Applied Probability , 51(2):472–495, 2019

    work page 2019

  6. [5]

    R. N. Bhattacharya and E. C. W aymire. A Basic Course in Probability Theory . Universitext - Springer-Verlag. Springer, 2007

    work page 2007

  7. [6]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular Variation. Encyclopedia of Mathematics and its Applications. Cambri dge University Press, 1987

    work page 1987

  8. [7]

    M. E. Caballero, J. L. P´ erez Garmendia, and G. Uribe Bravo . A Lamperti-type representation of continuous-state bran ching processes with immigration. The Annals of Probability , 41(3A):1585–1627, 2013. HARMONIC FUNCTIONS FOR KILLED MARKOV BRANCHING PROCESSES W ITH IMMIGRATION AND CULLING 19

    work page 2013

Show all 32 references
  1. [8]

    M. C. H. Choi and P. Patie. Skip-free Markov chains. Transactions of the American Mathematical Society , 371(10):7301–7342, 2019

    2019

  2. [9]

    K. L. Chung. Markov chains with stationary transition probabilities . Grundlehren der mathematischen Wissenschaften. Springe r, 1967

    1967

  3. [10]

    K. L. Chung and J. B. W alsh. Markov Processes, Brownian Motion, and Time Symmetry . Grundlehren der mathematischen Wissenschaften. Springer New York, 2006

    2006

  4. [11]

    R. A. Doney. A note on some results of Schuh. Journal of Applied Probability , 21(1):192–196, 1984

    1984

  5. [13]

    E. B. Dynkin. Markov Processes and Related Problems of Analysis . London Mathematical Society Lecture Note Series. Cambrid ge University Press, 1982

    1982

  6. [14]

    S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence . Wiley Series in Probability and Statistics. Wiley, 2009

    2009

  7. [15]

    Garcia-Millan, J

    R. Garcia-Millan, J. Pausch, B. W alter, and G. Pruessner . Field-theoretic approach to the universality of branchin g processes. Physical Review E , 98:062107, 2018

    2018

  8. [16]

    D. R. Grey. A note on explosiveness of Markov branching pr ocesses. Advances in Applied Probability , 21(1):226–228, 1989

    1989

  9. [17]

    T. E. Harris. The Theory of Branching Processes . Dover phoenix editions. Dover Publications, 2002

    2002

  10. [18]

    J. F. C. Kingman. Homecomings of Markov processes. Advances in Applied Probability , 5(1):66–102, 1973

    1973

  11. [19]

    Kuznetsov, A

    A. Kuznetsov, A. E. Kyprianou, and V. Rivero. The theory o f scale functions for spectrally negative L´ evy processes. In L´ evy Matters II: Recent Progress in Theory and Applications: Fra ctional L´ evy Fields, and Scale Functions , pages 97–186. Springer Berlin Heidelberg, B...

    2013

  12. [20]

    A. E. Kyprianou and Z. Palmowski. Fluctuations of spectr ally negative Markov additive processes. In C. Donati-Mart in, M. ´Emery, A. Rouault, and C. Stricker, editors, S´ eminaire de Probabilit´ es XLI, pages 121–135. Springer Berlin Heidelberg, Berlin, Heide lberg, 2008

    2008

  13. [21]

    A. Lambert. Population dynamics and random genealogies . Stochastic Models, 24(sup1):45–163, 2008

    2008

  14. [22]

    Li and Z

    B. Li and Z. Palmowski. Fluctuations of Omega-killed spe ctrally negative L´ evy processes. Stochastic Processes and their Appli- cations, 128(10):3273 – 3299, 2018

    2018

  15. [23]

    Li and A

    J. Li and A. G. Pakes. Asymptotic properties of the Markov branching process with immigration. Journal of Theoretical Proba- bility, 25:122–143, 2012

    2012

  16. [24]

    Z. Li. Measure-Valued Branching Markov Processes. Probability and Its Applications. Springer Berlin Heidel berg, 2010

    2010

  17. [25]

    P. W. Millar. A Path Decomposition for Markov Processes. The Annals of Probability , 6(2):345 – 348, 1978

    1978

  18. [26]

    J. R. Norris. Markov Chains . Cambridge Series in Statistical and Probabilistic Mathem atics. Cambridge University Press, 1998

    1998

  19. [27]

    K. R. Parthasarathy. Probability Measures on Metric Spaces . AMS Chelsea Publishing Series. Acad. Press, 1972

    1972

  20. [28]

    S. Redner. A Guide to First-Passage Processes . Cambridge University Press, 2001

    2001

  21. [29]

    L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales: Volume 2, Itˆ o Calculus . Cambridge Mathe- matical Library. Cambridge University Press, 2000

    2000

  22. [30]

    M. Vidmar. Exit problems for positive self-similar Mark ov processes with one-sided jumps. arXiv:1807.00486, 2018

    2018 arXiv

  23. [31]

    M. Vidmar. Fluctuation theory for upwards skip-free L´ e vy chains. Risks, 6(3):no. 102, 2018

    2018

  24. [32]

    W. W oess. Denumerable Markov Chains: Generating Functions, Boundar y Theory, Random Walks on Trees . EMS textbooks in mathematics. European Mathematical Society, 2009. Department of Mathematics, F aculty of Mathematics and Phys ics, University of Ljubljana Email address : mat...

    2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.