Killed path-dependent McKean-Vlasov SDEs for a probabilistic representation of non-conservative McKean PDEs

Daniela Morale; Leonardo Tarquini; Stefania Ugolini

REVIEW 4 major objections 5 minor 24 references

Killed path-dependent McKean-Vlasov SDEs for a probabilistic representation of non-conservative McKean PDEs

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Killed path-dependent McKean-Vlasov representation of a non-conservative PDE: plausible core, real and repairable proof gaps, worth a serious referee. read the letter →

arxiv 2507.23353 v1 pith:GNKTY4IZ submitted 2025-07-31 math.PR

Daniela Morale , Leonardo Tarquini , Stefania Ugolini This is my paper · ORCID

classification math.PR MSC 60H1060H3060K3560J6060J7560J8582C2282C31

keywords McKean-VlasovSDEkilledprocessnon-conservativereaction-diffusionPDEpath-dependentcoefficientsinteractingparticlesystemFeynman-Kacrepresentationsub-probabilitymeasuremarblesulphation

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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].

Load-bearing premise

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.

Editorial extensions

If this is right

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).

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

The 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.

Overall: 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.

Referee Report

4 major / 5 minor

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].

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 (4)

[Section 2.1, eq. (17)] 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 2.2, Proposition 2.7] 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 3, Theorem 3.1] 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.
[Equation (4) and following paragraph] 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.

minor comments (5)

[Section 2, eq. (5)] 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.
[Proposition 2.7, exponential estimate] 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 5, eq. (38)] 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).
[Definition 2.1] 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.
[Notation throughout Section 2] 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).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the killed-SDE/PDE representation is derived by Itô calculus from an independently defined McKean–Vlasov equation, and the [21] self-citations supply explicit coefficient estimates rather than the target result.

full rationale

The central claim is not circular by construction. The sub-probability law ν_t is defined as the law of the killed SDE (5),(7), not as the solution of PDE (1),(4); the identity (8)/(12) is a McKean–Vlasov fixed-point equation whose well-posedness is proved in Section 2 via the lifted system (19)–(20), and Theorem 3.2 derives the weak PDE formulation from Itô's formula and the compensator of the killing Poisson process. This is a genuine derivation, not a restatement of the input: nothing in the definition of the SDE or the stopping time presupposes the PDE solution. The paper does rely on the same authors' earlier work [21] for Proposition 2.2 and for the uniqueness-in-law proof pattern, but those citations concern elementary boundedness/Lipschitz estimates of the explicit coefficient b and a standard weak-uniqueness argument; they do not assume the killed-SDE well-posedness or the PDE representation as premises, so under the stated rules they are independent support rather than circular self-citation. The sign flaw in (17) — asserting 0 ≤ ∇K*ρ(·,x)(t) although ∇K integrates to zero and changes sign — is a substantive mathematical gap in a load-bearing estimate, but it is not a circular identification: the paper does not define ∇K*ρ as nonnegative or fit it to the conclusion it is used to prove. In short, no step reduces to its own input by construction or by a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is an analytical existence-uniqueness theorem. The model parameters (λ, c0, φ0, φ1), the kernel K, and the initial law are inputs from the model and are not fitted in this paper. The main postulated input is the uniform lower and upper bound on the denominator of b. No new physical entities are introduced; the cemetery state and the exponential clock are standard mathematical conventions.

assumptions (5)

domain assumption The denominator φ0 + φ1 c0 exp(-λ ρ(·,x)(t)) is bounded below by m and above by M for all t,x, where ρ solves (1),(4).
Asserted after (4); used in Prop. 2.2 and 2.3 to give b boundedness and Lipschitz continuity, which are the load-bearing regularity inputs for the SDE and particle system proofs.
domain assumption K is a smooth mollifier satisfying Definition 2.1: smooth, bounded first and second derivatives, Lipschitz K and ∇K, and integrates to 1.
Used throughout to control convolutions via (13)-(15) and to obtain the bounds (17), (22), and (23).
domain assumption Z is Exp(1) and independent of X and W.
Used in (7)-(9) to compute the survival probability and to define ν_t as the sub-probability law of the killed process.
domain assumption The initial data satisfy ρ0 in L2(R) ∩ Cb(R), E|ζ0|^2 < ∞, and ν0 = ρ0 dx.
Stated in (6); needed for the density argument and for SDE well-posedness.
standard math Standard theorems in stochastic analysis are used as black boxes: Girsanov with Novikov, Riesz representation, Grönwall's lemma, Yamada-Watanabe, and Wasserstein-space fixed point.
Invoked in the proofs of Prop. 2.7, Theorem 3.1, and Theorem 3.2.

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Pith. "Pith review of Killed path-dependent McKean-Vlasov SDEs for a probabilistic representation of non-conservative McKean PDEs." pith.science (2026). https://pith.science/paper/GNKTY4IZ

@misc{pith2026250723353,
  author       = {Pith},
  title        = {Pith review of: Killed path-dependent McKean-Vlasov SDEs for a probabilistic representation of non-conservative McKean PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GNKTY4IZ}},
  note         = {Machine review of arXiv:2507.23353}
}

read the original abstract

A McKean-Vlasov stochastic differential equation subject to killing associated to a regularised non-conservative and path-dependent nonlinear parabolic partial differential equation is studied. The existence and pathwise uniqueness of a strong solution and the regularity properties of its sub-probability law are proved. The density of such a law may be seen as a weak solution of the considered PDE. The well-posedness of the associated particle system is also discussed.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [21]

A Feynman--Kac representation of a non-conservative and path-dependent nonlinear reaction-diffusion-advection system

D. Morale, L. Tarquini, and S. Ugolini. A probabilistic interpretation of a non-conservative and path-dependent nonlinear reaction-diffusion system for the marble sulphation in Cul- tural Heritage. 2024. arXiv: 2407.19301 [math.PR]

work page Pith review arXiv 2024

[2] [1]

A mathematical model of sulphite chemical aggression of limestones with high permeability. Part I. Modeling and qualitative analysis

G. Al ` ı, V. Furuholt, R. Natalini, and I. Torcicollo. “A mathematical model of sulphite chemical aggression of limestones with high permeability. Part I. Modeling and qualitative analysis”. In: Transport in Porous Media 69.1 (2007), pp. 109–122

work page 2007

[3] [2]

A mathematical model of sulphite chemical aggression of limestones with high permeability. Part II. Numerical approxima- tion, Transport in Porous Media

G. Al ` ı, V. Furuholt, R. Natalini, and I. Torcicollo. “A mathematical model of sulphite chemical aggression of limestones with high permeability. Part II. Numerical approxima- tion, Transport in Porous Media”. In: Transport in Porous Media 69.1 (2007), pp. 175– 188

work page 2007

[4] [3]

Randomness in a nonlinear model of sulphation phenomena

F. Arceci, L.M. Giordano, M. Maurelli, D. Morale, and S. Ugolini. “Randomness in a nonlinear model of sulphation phenomena”. In: MACH2021. Mathematcal Modeling in Cultural Heritage. Ed. by G. Bretti, C. Cavaterra, M. Solci, and M. Spagnuolo. Springer Nature, 2023

work page 2023

[5] [4]

A reaction diffusion model with a stochastic boundary condition

F. Arceci, M. Maurelli, D. Morale, and S. Ugolini. “A reaction diffusion model with a stochastic boundary condition”. In: Mathematical Modeling in Cultural Heritage . Ed. by G. Bretti, C. Cavaterra, M. Solci, and M. Spagnuolo. Springer Nature Singapore, 2025, pp. 1–16

work page 2025

[6] [5]

Arceci, D

F. Arceci, D. Morale, and S. Ugolini. A numerical study of a PDE-ODE system with a stochastic dynamical boundary condition: a nonlinear model for sulphation phenomena

work page

[7] [6]

Discrete reaction-diffusion system with stochastic dynamical boundary conditions: convergence results

Francesca Arceci, Francesco Carlo De Vecchi, Daniela Morale, and Stefania Ugolini. Dis- crete reaction-diffusion systems with stochastic dynamical boundary conditions: conver- gence results. 2025. arXiv: 2507.09278

work page Pith review arXiv 2025

[8] [7]

A Mathematical Model for the Sulphur Dioxide Aggression to Calcium Carbonate Stones: Numerical Approximation and Asymp- totic Analysis

D. Aregba-Driollet, F. Diele, and R. Natalini. “A Mathematical Model for the Sulphur Dioxide Aggression to Calcium Carbonate Stones: Numerical Approximation and Asymp- totic Analysis”. In: SIAM Journal on Applied Mathematics 64.5 (2004), pp. 1636–1667

work page 2004