{"id":"7048f860-c5fa-4d18-8ce9-07efe47bf7f7","arxiv_id":"1908.04714","paper_version":5,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit harmonic functions for killed continuous-time branching processes with immigration and culling produce Laplace transforms of downward first-passage and explosion times.","lead":"The paper gives exact formulas for the chance that a random population that grows, dies, and occasionally gains or loses one member first drops to zero, and for the time it explodes, if the process is killed at an independent random time. It is a pure mathematics contribution that may sharpen population and avalanche models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The displayed formula for Φ_q in Theorem 3.6(i) has the exponential sign reversed; as written the integral diverges in an allowed case, so the central explicit harmonic function needs correction.","rationale":"The paper's main deliverable is the explicit harmonic function Φ_q and the Laplace-transform formulas built on it. The condition φ_q<φ is properly caveated in the abstract, but a sign reversal in the displayed integrand would make Theorem 3.6(i) false even when that condition holds. The proof of Theorem 3.6 states that ω solves ω′=−γ_qω, while the displayed definition ω=exp{−∫_v^{φ_q}γ_q} gives ω′=γ_qω. The only consistent reading is that the intended exponent is exp{−∫_{φ_q}^{v}γ_q}. This looks like a correctable typographical error rather than a conceptual flaw, so I would not reject the paper; however, acceptance should be conditional on correcting the displayed formula and the later displays in Theorem 4.2 and Corollary 4.14 that inherit it. Once corrected, the reader's substantive assessment—explicit harmonic functions modulo the φ_q<φ restriction—stands.","tokens_in":22996,"tokens_out":21719,"duration_ms":220777,"concrete_test":"Analytic check: take μ=0, λ=1, p0=p2=1/2, so φ=1, φ_q=0, ρ(v)=(1−v)^2/2 and γ_q(v)=2q/(1−v)^2. Evaluate the integral displayed in Theorem 3.6(i) at x=0. The as-printed integrand q·2(1−v)^{−2} exp{2qv/(1−v)} diverges at v=1, proving that the displayed formula is false. Then replace the exponential by exp{−∫_{φ_q}^{v}γ_q} = exp{−2qv/(1−v)}; the resulting Φ_q(0) is 1, and integration by parts verifies Eq. (3.2) for all x∈N, confirming that the corrected exponent is the intended harmonic function.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Read literally, Theorem 3.6(i) defines Φ_q(x) = q∫_0^φ exp{−∫_v^{φ_q} γ_q(w)dw}/ρ(v) v^x dv. This contradicts the proof's own ODE. Setting A(v)=∫_v^{φ_q}γ_q gives A′=−γ_q, so ω=exp{−A} satisfies ω′=γ_qω, not the asserted ω′=−γ_qω; the solution should be exp{−∫_{φ_q}^{v}γ_q}, equivalently exp{∫_v^{φ_q}γ_q}. The error is visible in the simplest allowed case: with μ=0, p0=p2=1/2, λ=1 and q>0, we have φ_q=0, φ=1, γ_q(v)=2q/(1−v)^2, and the displayed integrand is q·2(1−v)^{−2} exp{2qv/(1−v)}, which diverges at v=1. Hence, as printed, the integral defining Φ_q is not finite. The corrected exponent gives a finite integral and satisfies (3.2); Remark 3.12 expresses the corrected formula. The same sign reversal appears in the Φ_{q,q̄} display in Corollary 4.14. This is a load-bearing internal inconsistency, independent of the acknowledged φ_q<φ restriction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23244,"tokens_out":11852,"duration_ms":117871,"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":[{"comment":"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.","section":"Theorem 3.6(i), proof of Theorem 3.6"},{"comment":"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.","section":"Theorem 3.6, Ψ_q display"},{"comment":"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.","section":"Corollary 4.14"}],"minor_comments":[{"comment":"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.","section":"Section 3, Definition 3.5 vs. Theorem 3.6"},{"comment":"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.","section":"Proof of Theorem 3.6"},{"comment":"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.","section":"Remark 4.3"}],"recommendation":"major_revision","confidential_remarks":"The sign reversal in Theorem 3.6 and Corollary 4.14 appears to be a fixable typographical-level error rather than a conceptual flaw: the intended formulas are visible in Remark 3.12 and in the ODE derived before the theorem. Still, because the erroneous displays make the central harmonic functions divergent in an allowed case, the paper should not be accepted in its present form. I would be willing to re-review a corrected version. The limitation to q sufficiently large (φ_q<ϕ) is transparently stated and is a strength of the paper's honesty rather than a defect."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front.\n\nThe paper does something genuinely useful: it produces explicit harmonic functions for the q-killed continuous-time Bienaymé-Galton-Watson process with immigration and culling, and turns them into Laplace transforms of the first-passage-downward and explosion times. The comparison with Avram–Patie–Wang [3] and Duhalde–Foucart–Ma [12] is fair, and the explosion-time part plus the change-of-measure corollaries go beyond those papers. The abstract's caveat—this works only for killing rates with φ_q < φ, which always holds unless branching is supercritical with culling—is stated honestly and handled properly in Remark 3.8.\n\nThe problem: the displayed formula for Φ_q in Theorem 3.6(i) has the exponential sign reversed. The paper defines Φ_q with the factor exp{−∫_v^{φ_q} γ_q}, which is the reciprocal of what the generator equation (3.2) requires; the proof's own ω solves ω′ = −γ_q ω, i.e. it should be exp{−∫_{φ_q}^v γ_q}. As printed, the integral defining Φ_q diverges in perfectly allowed cases: take μ=0, p_0=p_2=1/2, λ=1, q>0 (critical binary branching, φ_q=0<ϕ=1); the integrand is 2q(1−v)^{-2} exp{2qv/(1−v)}, which blows up at v=1. So Theorem 3.6(i) as stated is false, and Definition 4.1, Theorem 4.2, and Corollary 4.14 all inherit the wrong sign.\n\nThat said, the intended mathematics is plainly right. Remark 3.12's μ=0 formula is the corrected version, and I checked the corrected Φ_q directly against (3.2): it is harmonic, bounded, vanishes at infinity, and gives correct limits, e.g. a.s. extinction for subcritical branching with immigration. This looks like a systematic display typo with a large blast radius, not a broken argument. Still, it is load-bearing, because explicit formulas are the paper's main product and the printed ones are not finite.\n\nWho this is for: specialists in branching processes and fluctuation theory. It deserves a serious referee, but only with the clear expectation that the sign be fixed and every downstream display rechecked. A first-pass report that found no critical red flags is too generous on the printed formulas. I would not cite the current arXiv version.","headline":"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.","tokens_in":23803,"tokens_out":39533,"would_cite":false,"duration_ms":322466,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J80","60J50"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["continuous-time Bienaymé-Galton-Watson process","branching with immigration and culling","killed Markov branching process","harmonic function","first passage downwards","explosion time","Laplace transform","skip-free Markov chain"],"falsifier":"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.","tokens_in":22762,"feed_emoji":"⏳","tokens_out":6262,"duration_ms":60116,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact Laplace transforms found for killed branching populations","feed_subtitle":"Closed-form harmonic functions give downward hitting and explosion transforms once the killing rate exceeds a culling threshold.","key_machinery":"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 $f(x)/f(a)$.","core_discovery":"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 $\\varphi\\le\\varphi$.","pith_inferences":["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."],"forward_implications":["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 $\\Phi_q$."],"supporting_citations":[{"why":"Supplies the prior first-passage treatment for continuous-time branching without immigration/culling, whose excessive-function approach this paper refines and makes explicit.","marker":"[3]"},{"why":"Gives the analogous explicit first-passage formula for continuous-state branching with immigration, the scaling-limit counterpart that formula (4.6) is designed to match.","marker":"[12]"},{"why":"Provides the scale-function paradigm for upwards skip-free random walks that motivates viewing $\\Phi_q$ as the minimal scale function.","marker":"[4]"},{"why":"Underlies Lemma 3.4: the root $\\varphi_q$ of $q=\\mu(\\tilde r(z)-1)$ comes from fluctuation theory for Poisson-subordinated left-continuous random walks.","marker":"[31]"},{"why":"Source for the classical branching-process facts, including the explosivity criterion used to handle the explosion part of the results.","marker":"[17]"},{"why":"Gives the Laplace-transform uniqueness theorem used to conclude that the transforms on a neighbourhood of infinity determine the laws.","marker":"[5]"},{"why":"Provides the general path decomposition at the minimum that Proposition 5.5 and Corollary 5.7 specialize to the present setting.","marker":"[25]"}],"fun_headline_variants":["Exact killing-time transforms for branching processes with culling","Harmonic functions yield closed-form Laplace transforms for hits","First-passage Laplace transforms via explicit harmonic functions","Branching process killing: explicit formulas for passage and explosion","Closed-form harmonic functions decode branching hitting and explosion times"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact killing-time transforms for branching processes with culling","Harmonic functions yield closed-form Laplace transforms for hits","First-passage Laplace transforms via explicit harmonic functions","Branching process killing: explicit formulas for passage and explosion","Closed-form harmonic functions decode branching hitting and explosion times"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":3031,"prompt_tokens":982,"completion_tokens":2049,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":1972}},"tokens_in":598,"tokens_out":2049,"duration_ms":15598,"temperature":1.0,"reasoning_tokens":1972,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:35:00.502880+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Avram, P","cited_arxiv_id":null,"evidence_quote":"Supplies the prior first-passage treatment for continuous-time branching without immigration/culling, whose excessive-function approach this paper refines and makes explicit."},{"cited_title":"Duhalde, C","cited_arxiv_id":null,"evidence_quote":"Gives the analogous explicit first-passage formula for continuous-state branching with immigration, the scaling-limit counterpart that formula (4.6) is designed to match."},{"cited_title":"Avram and M","cited_arxiv_id":null,"evidence_quote":"Provides the scale-function paradigm for upwards skip-free random walks that motivates viewing $\\Phi_q$ as the minimal scale function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underlies Lemma 3.4: the root $\\varphi_q$ of $q=\\mu(\\tilde r(z)-1)$ comes from fluctuation theory for Poisson-subordinated left-continuous random walks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source for the classical branching-process facts, including the explosivity criterion used to handle the explosion part of the results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Laplace-transform uniqueness theorem used to conclude that the transforms on a neighbourhood of infinity determine the laws."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the general path decomposition at the minimum that Proposition 5.5 and Corollary 5.7 specialize to the present setting."}],"review_version":1}