{"id":"3218e981-832b-48cb-9f04-ecbac30a144e","arxiv_id":"2507.23353","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A killed McKean-Vlasov SDE is shown to be well posed, and the density of its sub-probability law is a weak solution of the associated non-conservative path-dependent McKean PDE.","lead":"This paper proves that a random process which can be killed at a random time gives a probabilistic picture of a nonlinear reaction-diffusion equation where particles disappear, not just move. It matters because it supplies rigorous microscopic dynamics for a marble-sulfation model and for a broader class of non-conservative McKean PDEs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's assertion 0≤∇K*ρ in (17) is false for any smooth mollifier, and the Lipschitz estimates in Propositions 2.3, 2.7, and 4.1 depend on it.","rationale":"The reader's weakest assumption, the uniform lower bound m > 0 on the denominator in (4), is real but is an explicit hypothesis; if the parameters satisfy it, the boundedness arguments go through. The nonnegativity of ∇K*ρ in (17) is a different and more concrete problem: it is not an extra assumption but an asserted inequality that is impossible for any nonconstant C^1 probability kernel. Since the proof of Proposition 2.3 and the applications of Proposition 2.2 in Proposition 2.7 both require the second argument of b to lie in R+0, the false lower bound is load-bearing for the well-posedness results that underlie the PDE representation. The same issue affects the particle system proof in Proposition 4.1. The concern does not show the central claims are false: b is in fact Lipschitz on the two-sided box of possible arguments, so the proofs are likely repairable by replacing the false lower bound with an absolute-value bound and extending the domain in Proposition 2.2. For this reason I keep the reader's CONDITIONAL verdict and do not move to REJECT or ACCEPT; the open condition is that the repair is carried out and the affected estimates are rechecked. The reader's rationale already lists the unsupported nonnegativity of ∇K*ρ as one of the gaps, but their single weakest-assumption slot is occupied by the denominator bound, so my agreement is partial.","tokens_in":17075,"tokens_out":10718,"duration_ms":127964,"concrete_test":"Take K to be the standard Gaussian density and ρ0(dy) = δ_{y0}(dy) with y0 > 0. For x > y0, ∇K(x−y0) < 0, so for small t > 0, ∫_0^t ∇K*ρ(s,x) ds < 0, contradicting the lower bound in (17). Then check whether the proof of Proposition 2.7 still goes through when (17) is replaced by |∇K*ρ| ≤ M'_K T and Proposition 2.2 is extended to y,y′ ∈ [−M'_K T, M'_K T]; if the two-sided Lipschitz estimate for b holds, the concern is a repairable proof gap rather than a failure of the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Central well-posedness rests on boundedness and Lipschitz continuity of the drift b in (4). Proposition 2.3 proves Lipschitz continuity by first claiming in (17) that 0 ≤ ∇K*ρ(·,x)(t) ≤ M'_K T. For a C^1 probability kernel K, ∫_R ∇K(z) dz = K(∞) − K(−∞) = 0, and ∇K is not identically zero, so ∇K takes negative values. For ρ with mass in a region where K'(x−y) < 0, ∇K*ρ(·,x)(t) < 0; hence (17) is false as written. The false lower bound is not cosmetic: it is what puts the second argument y = ∇K*ρ into the domain R+0 required by Proposition 2.2, and Proposition 2.7 uses exactly that local Lipschitz estimate to obtain the Wasserstein contraction leading to pathwise uniqueness. Proposition 4.1 inherits the same requirement. The argument can likely be repaired by using |∇K*ρ| ≤ M'_K T and extending the elementary Lipschitz bound for b to the two-sided box x ∈ [0, M_KT], y ∈ [−M'_K T, M'_K T]; the derivative ∂_y b is bounded there because e^{−λxy} ≤ e^{λ M_K M'_K T^2}. But as written the proof has a false premise at a load-bearing point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a killed McKean-Vlasov SDE whose coefficients depend on the time-integrated, spatially mollified law of the surviving process, with a killing time governed by a state-dependent exponential intensity. The main results are: pathwise strong existence and uniqueness for the killed SDE (Theorem 2.8, via the auxiliary lifted system in Proposition 2.7); existence of an L^p density for the sub-probability law and identification of that density as a weak solution of the non-conservative path-dependent PDE (Theorems 3.1 and 3.2); and well-posedness of the associated finite particle system (Proposition 4.1). The coefficient class is the specific family (4), and the paper positions itself as a probabilistic complement to the authors' earlier Feynman-Kac representation in [21].","tokens_in":17315,"tokens_out":12843,"duration_ms":147184,"significance":"If the proofs are completed, the paper would give a natural killed-process representation of a non-conservative, path-dependent reaction-diffusion PDE, and it would unify the strong-solution and density approaches with the earlier analytic representation. The construction of the killing time through an inhomogeneous Poisson time-change and the lift to an R^2-valued system are clean ideas, and the particle-system well-posedness is a useful complement. The paper also states clear boundedness and Lipschitz conditions under which the main results are claimed to extend to general coefficients. However, several load-bearing estimates are incomplete or false as written: the sign bound (17), the contraction argument in Proposition 2.7, and the Girsanov step in Theorem 3.1 all need repair before the central claims are supported. The uniform lower bound on the denominator in (4) is also an unproved assumption on the data.","major_comments":[{"comment":"The bound 0 ≤ ∇K*ρ(·,x)(t) ≤ M'_K T is false for any C^1 probability kernel K, because ∫_R ∇K(z) dz = 0 and ∇K changes sign; for the Gaussian kernel, for example, the convolution can be negative where ρ has mass on one side of x. This is not cosmetic: Proposition 2.3 applies Proposition 2.2 with y and y' assumed to lie in R_+^0, and Proposition 2.7 uses the same nonnegativity through (22)-(23). The argument can likely be repaired by replacing the false lower bound with |∇K*ρ(·,x)(t)| ≤ M'_K T and proving the Lipschitz estimate for b on the symmetric box x ∈ [0, M_K T], y ∈ [-M'_K T, M'_K T], where ∂_y b is bounded by a constant depending on exp(λ M_K M'_K T^2). As written, the proof of Proposition 2.3 rests on a false premise, and Proposition 4.1 inherits the same issue when it invokes Proposition 2.3 for the empirical density.","section":"Section 2.1, eq. (17)"},{"comment":"The contraction argument for existence is incomplete. The equality D_T^2(Θ(eµ), Θ(eµ')) = D_T^2(eµ, eµ') in (28) is asserted before eµ and eµ' are known to be fixed points of Θ; the iteration argument requires instead a stability estimate for D_T(Θ(m), Θ(n)) for arbitrary input measures m and n, which is not derived. Moreover, Lemma 2.6 is applied to D_t^2(eµ, eµ') with eµ, eµ' laws on R^2, while the bound in (25) controls only |eX_a - eX'_a| and not the displacement of the eΛ-coordinate; without an additional estimate for the second component, inequality (26) does not follow. These gaps propagate to Theorem 2.8, whose proof is presented as a direct consequence of Proposition 2.7.","section":"Section 2.2, Proposition 2.7"},{"comment":"The proof of density existence is not valid as written. The inequality H_t(f) ≤ E[f(eX_t)] is asserted for arbitrary f ∈ C_c^∞, which is false for signed f; the correct starting point is |H_t(f)| ≤ E[|f(eX_t)|]. In addition, equation (31) expresses E_P[f(eX_t)] as E_P[f(X_0 + √2 W_t)(Z_T)^{-1}], but the Girsanov transformation with the displayed density gives, on the path-space level, an expectation of f at the Brownian motion under Q multiplied by Z_T^{-1}; the reduction to a simple product of f(X_0 + √2 W_t) with (Z_T)^{-1} is not justified and appears to have the wrong density direction. The L^q bound may be recoverable by a different argument, but as written the continuity estimate (30) for H_t is not established.","section":"Section 3, Theorem 3.1"},{"comment":"The uniform lower bound m > 0 for φ0 + φ1 c0 exp(-λρ(·,x)(t)) is assumed for the PDE solution, but no explicit conditions on φ0, φ1, c0, ρ0, and T are given that guarantee it; if φ1 is negative and the denominator approaches zero along the evolution, the drift b is unbounded and Proposition 2.2 fails. This premise is load-bearing because boundedness and Lipschitz continuity of b are used in Propositions 2.3, 2.7, and 4.1. Since Theorem 3.2 constructs v only later, the paper cannot simply assert the bound a priori; it needs either explicit parameter hypotheses ensuring m > 0 or a proof that the measure constructed by the SDE keeps the denominator uniformly positive.","section":"Equation (4) and following paragraph"}],"minor_comments":[{"comment":"The notation I(τ,T) := [0,τ) ∪ [0,T] is tautological and equals [0,T]; the intended interval is probably [0,τ) ∩ [0,T] or a similar convention, as used in Section 4.","section":"Section 2, eq. (5)"},{"comment":"In bounding A1, the proof replaces exp(|y_u - y'_u|) by 1 + |y_u - y'_u| without an exponential factor; since |y_u - y'_u| ≤ λc0T, the correct bound has an additional factor e^{λc0T}. This is a missing constant rather than a conceptual error.","section":"Proposition 2.7, exponential estimate"},{"comment":"The reaction term in the unregularized model appears to be missing the exponential: the intended term is λc0 e^{-λρ(·,x)(t)} ρ(t,x), not λc0(-λρ(·,x)(t))ρ(t,x).","section":"Section 5, eq. (38)"},{"comment":"The definition of a smooth mollifier requires boundedness and Lipschitz continuity of K and ∇K but does not require ∇K ∈ L^1(R); since K is a probability density, one can assume integrability of ∇K if needed for the convolution estimates, and this should be stated explicitly.","section":"Definition 2.1"},{"comment":"The process eX is sometimes one-dimensional and sometimes the two-dimensional pair (eX, eΛ); for instance, in Proposition 2.7 the map Θ is said to satisfy Θ(eµ) = L(eX) where eX is the R^2-valued solution. The notation should be made consistent, for example by writing X = (X, Λ).","section":"Notation throughout Section 2"}],"recommendation":"major_revision","confidential_remarks":"The paper leans heavily on the authors' earlier preprint [21] for Proposition 2.2 and for one uniqueness-in-law step; the editor may wish to ensure that the cited statements in [21] are publicly available in final form and that the assumptions match those used here. The scope fits a probability journal, and the killed-process perspective is a genuine contribution, but the proof gaps listed in the major comments should be addressed before the paper is accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper is worth reading, but I would not trust the theorems as printed. The central construction—a McKean-Vlasov SDE killed at a path-dependent, state-dependent rate that represents a non-conservative reaction-diffusion PDE—is sensible, and the authors are honest that it is an equivalent, more probabilistic alternative to their own Feynman-Kac formulation in [21]. The PDE representation via Itô's formula and the finite particle system are natural. I believe the conclusions are likely true; the gaps are technical, not fatal, and most are repairable.\n\nWhat is new: the killed SDE (5),(7), the lifted system with killing only in the law, and the well-posedness result for the particle system. The paper also does a decent job of explaining why the survival probability is e^{-Λ_t} and of connecting the present formulation to the earlier McKean-Feynman-Kac framework. The reliance on [21] is not a red flag; the earlier paper covers the Feynman-Kac side, and this is a genuine complement.\n\nThe soft spots are real. The most concrete is in Proposition 2.3: the bound (17) asserts 0 ≤ ∇K*ρ(·,x)(t). For any smooth mollifier K, ∫∇K = 0 and ∇K is not identically zero, so ∇K*ρ can be negative as soon as the mass of ρ overlaps the negative part of K'. That lower bound is load-bearing: it is what puts the second argument of b into the positive quadrant required by Proposition 2.2, and the Wasserstein contraction in Proposition 2.7 inherits this. The fix is straightforward—use |∇K*ρ| ≤ M'_K T and extend the Lipschitz and boundedness estimates for b to a two-sided box, where e^{-λxy} is bounded by e^{λ M_K M'_K T^2}. But as written, the proof rests on a false premise at a central point. The contraction argument in Prop 2.7 also has a circular step: inequality (26) is derived for two solutions, i.e., fixed points of Θ, and is then used to prove Θ is a contraction before the fixed point is known. Repairable, but not just a missing constant. Theorem 3.1's step H_t(f) ≤ E[f(eX_t)] is false for signed f (the killing factor e^{-Λ}≤1 flips the inequality for negative f), though the absolute-value version is what is needed. Proposition 4.1 silently extends Prop 2.3 from densities to empirical sub-probability measures; that extension is true but unstated. Also, the assumed lower bound m>0 on φ0+φ1 c0 e^{-λρ(·,x)(t)} does not obviously control the denominator of b, whose first argument is K*ρ(·,x)(t), not ρ(·,x)(t). That mismatch deserves a close look.\n\nOverall: this is a serious paper with a plausible core, but it needs a major revision. I would send it to a referee with instructions to focus on the sign bounds, the fixed-point argument, and the denominator assumption. I would not cite it in its current form.","headline":"Killed path-dependent McKean-Vlasov representation of a non-conservative PDE: plausible core, real and repairable proof gaps, worth a serious referee.","tokens_in":17907,"tokens_out":7870,"would_cite":false,"duration_ms":81798,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60H30","60K35","60J60","60J75","60J85","82C22","82C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the survivor law of a killed McKean–Vlasov diffusion is a weak solution of a non-conservative, path-dependent reaction-diffusion PDE.","keywords":["McKean-Vlasov SDE","killed process","non-conservative reaction-diffusion PDE","path-dependent coefficients","interacting particle system","Feynman-Kac representation","sub-probability measure","marble sulphation"],"falsifier":"Simulate the killed particle system with a Gaussian mollifier, form the empirical density of the survivors as $N$ grows, and evaluate the residual in the weak formulation of Theorem 3.2 for several smooth test functions: a residual that does not vanish as $N\\to\\infty$ would refute the representation. A preliminary probe is to track $\\varphi_0 + \\varphi_1 c_0 \\exp(-\\lambda \\int_0^t K*\\rho(\\cdot,x)(s)\\,ds)$: if it reaches zero before $T$, the boundedness premise behind the existence theorems has failed.","tokens_in":16813,"feed_emoji":"🧪","tokens_out":16267,"duration_ms":169543,"temperature":0.7,"pith_summary":"This paper aims to establish a probabilistic representation for a class of nonlinear reaction-diffusion equations in which mass is lost over time and the coefficients look at the entire past of the density. The representation is a killed McKean–Vlasov diffusion: a Brownian particle pushed by a mean-field drift and stopped at a random time whose intensity depends on the mollified density accumulated over time. The paper proves that the law of the surviving particles has a density and that this density is a weak solution of the PDE; it also proves well-posedness of the one-particle equation and of the associated $N$-particle system. This matters because it turns a non-conservative, path-dependent PDE into a concrete particle model of the same class that appears in marble-sulphation modelling, and it gives the PDE an existence proof through stochastic analysis.","feed_headline":"Killed particle paths solve a mass-losing reaction-diffusion PDE","feed_subtitle":"The law of surviving particles is shown to be a weak solution, giving this PDE a well-posed particle basis.","key_machinery":"Two devices carry the argument. The first is the representation of the killing time as the first jump of a time-changed Poisson process with compensator $\\Lambda_t = \\int_0^t \\lambda c_0 \\exp(-\\lambda \\int_0^s K*\\nu_r(X_s)\\,dr)\\,ds$; this places the reaction term inside Itô's formula as a jump term. The second is the lifted process $(X,\\Lambda)$ together with the map $\\Phi(\\mu)(dx)=\\int_{\\mathbb{R}^2} e^{-y} \\mu(dx,dy)$, which converts sub-probability measures on $\\mathbb{R}$ into genuine probability measures on $\\mathbb{R}^2$. The lifted formulation turns the non-conservative problem into a standard McKean–Vlasov fixed point: bounding the Wasserstein distance $D^2_t$ between two solutions yields a Gronwall inequality, hence pathwise uniqueness, and Yamada–Watanabe upgrades this to strong existence. The uniform bounds $m$ and $M$ on the denominator in the drift coefficients are what keep those coefficients bounded and Lipschitz, so the contraction estimates close.","core_discovery":"At its core, the paper establishes that the non-conservative, path-dependent reaction-diffusion PDE (1),(4) is the macroscopic image of a killed McKean–Vlasov diffusion. If $X$ solves (5) with killing time (7), then $\\nu_t := P(X_t \\in \\cdot, t<\\tau)$ is a sub-probability measure, and it has a density $v(t,\\cdot)$. Theorem 3.2 shows that this density satisfies the weak formulation of the PDE, displayed as the identity between $\\int f\\,d\\nu_t$ and the sum of the initial term, the diffusion term, the drift term, and the reaction term $-\\int_0^t \\int f\\, \\lambda c_0 e^{-\\lambda K*v(\\cdot,x)(s)} v(s,x)\\,dx\\,ds$. Theorem 2.8 gives a strong, pathwise unique solution of the killed equation, and Proposition 4.1 gives the same well-posedness for the finite system of surviving particles. The construction is offered as the microscopic counterpart of the authors' earlier Feynman–Kac representation of the same PDE in [21].","pith_inferences":["A natural testable extension is the singular limit in which the mollifier width tends to zero: the authors list this as future work, and the natural conjecture is that the killed-particle laws converge to a weak solution of the unregularized sulphation PDE (38).","The SDE uniqueness proof suggests a route to PDE uniqueness that the paper itself does not take: two weak solutions arising as densities of killed-diffusion laws would have to coincide by the Wasserstein contraction in Proposition 2.7.","The standing assumption that the drift denominator stays bounded away from zero can be probed numerically for negative $\\varphi_1$ and concentrated initial data; finding parameters where it hits zero would mark the boundary of the theory and predict a finite-time breakdown of the particle model."],"forward_implications":["A direct corollary of Theorem 3.2 is existence of a weak solution of the non-conservative, path-dependent PDE (1),(4), exhibited as the marginal density of the killed diffusion rather than obtained by PDE methods.","Theorem 2.8 makes the killed McKean–Vlasov SDE a legitimate object for simulation: pathwise uniqueness in law means numerical schemes for (5),(7) have a well-defined target.","Proposition 4.1 validates the finite particle system of survivors as a well-posed approximation of the same sub-probability law, so the empirical measure of alive particles is a principled Monte Carlo estimator of the PDE solution.","Because the proofs use only boundedness and Lipschitz continuity of the coefficients, the representation extends to any advection and reaction terms with those properties, not only the explicit exponential coefficients (4)."],"supporting_citations":[{"why":"Prior Feynman–Kac representation of the same PDE; its Propositions 3.1–3.2 supply the boundedness and Lipschitz estimates for the drift used throughout Sections 2 and 4.","marker":"[21]"},{"why":"Provides the probabilistic representation method for non-conservative nonlinear PDEs and the uniqueness-in-law comparison that the paper adapts to the path-dependent case.","marker":"[17]"},{"why":"Supplies the map from probabilities on R^2 to sub-probability measures on R (Proposition 3.12) and the constructive particle-system well-posedness scheme (Section 3.1) used in Sections 2 and 4.","marker":"[13]"},{"why":"Cited jointly in Theorem 3.1 as the source of the technique used to prove that the sub-probability law has a density.","marker":"[19, 23]"},{"why":"Yamada–Watanabe theorem from this reference upgrades weak existence plus pathwise uniqueness to strong well-posedness.","marker":"[16, p. 308]"},{"why":"Provides the Wasserstein-distance properties (Chapter 6) used in Lemma 2.6 and in the contraction estimates.","marker":"[24]"},{"why":"Riesz representation theorem used to turn the continuous linear functional into the density v(t,·) in L^p.","marker":"[10]"}],"fun_headline_variants":["Surviving particles encode non-conservative PDE solutions","Killed paths give weak solutions to non-conservative PDE","Survivor law is a weak solution to path-dependent PDE","Non-conservative PDE solved by killed particle paths"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the denominator in the drift staying bounded away from zero throughout [0,T]; if the parameters and the evolving density let it approach zero, the drift becomes unbounded and the existence and uniqueness proofs fall apart.","fun_headline_variants_meta":{"raw":{"variants":["Surviving particles encode non-conservative PDE solutions","Killed paths give weak solutions to non-conservative PDE","Survivor law is a weak solution to path-dependent PDE","Non-conservative PDE solved by killed particle paths"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000474,"raw_usage":{"total_tokens":2312,"prompt_tokens":861,"completion_tokens":1451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":1383}},"tokens_in":477,"tokens_out":1451,"duration_ms":11090,"temperature":1.0,"reasoning_tokens":1383,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:50:56.498877+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the killed particle system with a Gaussian mollifier, form the empirical density of the survivors as $N$ grows, and evaluate the residual in the weak formulation of Theorem 3.2 for several smooth test functions: a residual that does not vanish as $N\\to\\infty$ would refute the representation. A preliminary probe is to track $\\varphi_0 + \\varphi_1 c_0 \\exp(-\\lambda \\int_0^t K*\\rho(\\cdot,x)(s)\\,ds)$: if it reaches zero before $T$, the boundedness premise behind the existence theorems has failed.","supporting_citations":[{"cited_title":"A Feynman--Kac representation of a non-conservative and path-dependent nonlinear reaction-diffusion-advection system","cited_arxiv_id":"2407.19301","evidence_quote":"Prior Feynman–Kac representation of the same PDE; its Propositions 3.1–3.2 supply the boundedness and Lipschitz estimates for the drift used throughout Sections 2 and 4."},{"cited_title":"Probabilistic representation of a class of non conservative nonlinear Partial Differential Equations","cited_arxiv_id":null,"evidence_quote":"Provides the probabilistic representation method for non-conservative nonlinear PDEs and the uniqueness-in-law comparison that the paper adapts to the path-dependent case."},{"cited_title":"Hambly and P","cited_arxiv_id":null,"evidence_quote":"Supplies the map from probabilities on R^2 to sub-probability measures on R (Proposition 3.12) and the constructive particle-system well-posedness scheme (Section 3.1) used in Sections 2 and 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Wasserstein-distance properties (Chapter 6) used in Lemma 2.6 and in the contraction estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Riesz representation theorem used to turn the continuous linear functional into the density v(t,·) in L^p."}],"review_version":1}