{"id":"e89641ec-47d8-4fd7-b40e-d79973f352b3","arxiv_id":"2508.12979","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Leibenson equation is identified as a nonlinear Fokker-Planck equation and given a probabilistic representation by a McKean-Vlasov SDE whose solutions form the Leibenson process.","lead":"This paper shows that the Leibenson equation, a combined porous media and p-Laplace equation, can be written as a nonlinear Fokker-Planck equation and has a counterpart stochastic process called the Leibenson process. It proves that the Barenblatt solutions of this equation match the density curves of unique solutions to a novel McKean-Vlasov stochastic differential equation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Marginal regularity at the free boundary is the load-bearing assumption: the SDE coefficients involve ∂²ρ, and the abstract supplies no proof that this is finite on the Barenblatt support.","rationale":"The reader's weakest assumption identified the same core issue: the SDE coefficients require enough regularity of the marginal densities. My stress-test sharpens this by pointing to the free boundary of the Barenblatt profile, where derivative blow-up is a concrete possibility for degenerate nonlinear diffusion. However, because the full text is unavailable, I cannot determine whether the paper already proves the necessary regularity or restricts parameters. Thus the appropriate verdict remains unverified, exactly as the reader concluded. The proposed test would settle the concern by checking the explicit profile for a representative parameter pair, and if the derivatives are finite, the concern would be resolved. No change to the reader's verdict is needed; the concern only reinforces that the paper's central claim depends on a nontrivial regularity result that is not visible from the abstract.","tokens_in":745,"tokens_out":4727,"duration_ms":60476,"concrete_test":"Locate the explicit Barenblatt solution in the paper and compute, for a representative degenerate parameter pair (e.g., p>2, q<1), the second derivative of v(t,x)=u(t,x)^q as x approaches the boundary of the support. Check whether it is bounded and continuous. If it is unbounded, check whether the SDE coefficients are defined by some limiting convention or whether the theorem excludes that parameter range. If neither is the case, the strong-existence claim is not established for that range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts unique strong solutions to a McKean–Vlasov SDE whose coefficients depend pointwise on ρ_t(x), ∂_x ρ_t(x), and ∂_xx ρ_t(x). For this to be meaningful, these derivatives must exist and be evaluable at the points visited by the process. For the Barenblatt solutions of ∂_t u = Δ_p u^q, this is non-obvious at the free boundary: the pressure v = u^q behaves like a power of the distance to the boundary, and after taking the q-th root, the density u may have singular derivatives. In particular, ∂_xx v, which enters the coefficients, may be unbounded near the interface for some p>1, q>0. The abstract says only that the drift is of bounded variation, but BV drift does not control second derivatives of the density. The paper must therefore contain a regularity theorem for the marginals, or a parameter restriction, showing that these derivatives are bounded on the support. The abstract does not state such a result. If this regularity fails, the SDE is not even well-defined, and the claimed probabilistic representation collapses. This is the weakest link because it is a precondition for the entire construction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to identify the Leibenson equation ∂_t u = Δ_p u^q, a simultaneous generalization of the porous medium equation and the p-Laplace equation, as a nonlinear Fokker–Planck equation. It further claims to construct a McKean–Vlasov SDE whose coefficients depend pointwise on the time-marginal density and its first and second spatial derivatives, and to show that the Barenblatt solutions of the PDE are exactly the one-dimensional marginal densities of the unique solutions to this SDE. The resulting process, called the Leibenson process, is asserted to be a nonlinear Markov process. A further central claim is that, despite degenerate diffusion and a drift of bounded variation, these solutions are probabilistically strong.","tokens_in":997,"tokens_out":2234,"duration_ms":28613,"significance":"If the claimed results hold, the paper would provide a substantive bridge between nonlinear PDEs of Leibenson type and Markovian stochastic processes, extending known probabilistic representations for porous medium and p-Laplace equations to a broader family. The novelty of an SDE whose coefficients depend on derivatives of the marginal density, together with strong uniqueness under degenerate diffusion and irregular drift, would be a valuable contribution. However, the abstract alone provides no proofs or derivations, and the central regularity requirement for the marginal densities is not addressed. The full text may supply the necessary theorems, but they are not visible in the abstract.","major_comments":[{"comment":"The SDE coefficients depend pointwise on ρ_t(x), ∂_x ρ_t(x), and ∂_xx ρ_t(x). For the construction to be well-posed, these derivatives must exist and be finite at the points visited by the process. The abstract states only that the drift is of bounded variation, which does not control second derivatives of the density. This is particularly delicate at the free boundary of the Barenblatt solution, where the pressure v = u^q behaves like a power of the distance to the boundary; after taking the q-th root, the density u may have singular derivatives, and ∂_xx v entering the coefficients may be unbounded for some p > 1, q > 0. The manuscript must contain a regularity theorem for the marginal density, or a parameter restriction, proving that ∂_xx ρ is bounded (or at least integrable in a suitable sense) on the support. Without this, the SDE is not even well-defined, and the claimed representa","section":"Abstract, regularity of marginal densities"},{"comment":"The abstract asserts that the McKean–Vlasov SDE has unique solutions and that these solutions are probabilistically strong. Because the drift depends on the solution's marginal density and its derivatives, the usual theory of SDEs with BV drift does not directly apply; there is a circular dependence of the coefficients on the law, and the diffusion is degenerate. The manuscript must provide a precise definition of solution (weak vs. strong, pathwise uniqueness vs. uniqueness in law) and prove existence and uniqueness in that class. The abstract gives no indication of the method, but this is essential to the central claim.","section":"Abstract, uniqueness and strong existence"},{"comment":"The abstract states the equation is considered for p > 1 and q > 0, but the existence and form of Barenblatt solutions, as well as the required regularity of their marginals, may depend on additional restrictions on p and q. For the porous medium case (p = 2) the range q > 0 is natural, but for general p the regularity at the free boundary can differ. If the theorem does not cover all p > 1, q > 0, the abstract should state the exact parameter range. If it does cover all, the regularity claim becomes stronger and must be justified.","section":"Abstract, parameter range"}],"minor_comments":[{"comment":"The notation Δ_p f is defined, but the reader is not told how u^q is interpreted when u is nonnegative (which is presumably the relevant case for Barenblatt solutions). This should be clarified.","section":"Abstract, notation"},{"comment":"The abstract cites no prior work on probabilistic representations for porous medium or p-Laplace equations. A brief contextualization in the abstract or introduction would help position the novelty.","section":"Abstract, references"},{"comment":"This review is based on the abstract only; the full manuscript was not available. Consequently, my comments are limited to what can be inferred from the abstract, and I cannot verify the actual derivations or proofs.","section":"General"}],"recommendation":"uncertain","confidential_remarks":"I was asked to review this paper but only the abstract was provided. The central claim is plausible but rests on a regularity property of the marginal density at the free boundary that is not addressed in the abstract. Before any decision can be reached, the manuscript must be inspected for a proof of the boundedness/regularity of ∂_xx ρ on the support, or a parameter restriction that guarantees it. The 'uncertain' recommendation reflects lack of access to the full text, not a detected error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nTwo things up front. First, this review is based on the abstract only, so any verdict is provisional. Second, the central claim is genuinely new: a McKean–Vlasov SDE whose coefficients depend pointwise on the marginal density and its first and second derivatives, giving a probabilistic representation of Barenblatt solutions of the Leibenson equation. If it holds, it connects a broad class of nonlinear diffusion PDEs to stochastic analysis and to a nonlinear Markov process. That would be a real contribution.\n\nWhat the paper does well, on paper: it identifies the Leibenson equation as a nonlinear Fokker–Planck equation, constructs the associated McKean–Vlasov SDE, and proves existence/uniqueness of strong solutions despite a degenerate diffusion and a drift that is merely of bounded variation. The authors are well-known in this area, so the claim that they have a proof of strong well-posedness under those conditions is worth taking seriously.\n\nThe soft spot is precisely the one the stress-test flags. The coefficients involve ∂²ρ, so the SDE is only well-defined if the marginal densities are regular enough at the points the process visits. For Barenblatt solutions of ∂tu = Δ_p u^q, that means regularity at the free boundary. The pressure v = u^q behaves like a power of distance to the boundary; after taking the q-th root, u may develop singular derivatives, and ∂xx v enters the drift. BV drift does not control second derivatives of the density. The abstract states no regularity theorem and no parameter restrictions ensuring these derivatives are bounded on the support. This is not necessarily fatal—the full paper may contain exactly the needed estimate—but it is load-bearing. Without it, the representation is formal rather than probabilistic.\n\nBecause we only see the abstract, I cannot confirm or deny the proofs. That is the main limitation: absence of evidence, not detected error.\n\nThis paper is for stochastic analysts and nonlinear PDE people who work with McKean–Vlasov equations and degenerate diffusions. It deserves a serious referee. The right referee will check the regularity of marginals near the free boundary and the strong well-posedness argument. I would send it to peer review rather than desk reject, but I would not trust the conclusion until the full text is on the table.\n\nMy own verdict: unverdictable from the abstract, but clearly worth a careful look.","headline":"Novel McKean–Vlasov representation of the Leibenson equation with derivative-dependent coefficients; plausible and important, but abstract-only and the free-boundary regularity of the marginals is the load-bearing assumption.","tokens_in":1435,"tokens_out":1957,"would_cite":false,"duration_ms":20289,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60J60","35K55","35Q84"],"pacs":[],"model":"deepseek-v4-flash","headline":"Barenblatt solutions are the marginals of a single stochastic process.","keywords":["Leibenson equation","Barenblatt solutions","McKean-Vlasov SDE","nonlinear Fokker-Planck equation","p-Laplace equation","porous media equation","nonlinear Markov process","strong solutions"],"falsifier":"Choose a parameter regime of strong degeneracy where the Barenblatt solution has a non-smooth interface or cusp. If the first or second spatial derivative of the marginal density fails to be a well-defined pointwise function at the interface, the coefficient evaluation in the McKean-Vlasov SDE is undefined and the claimed representation cannot hold; one could look for a time where the simulated marginal's derivative diverges while the PDE's Barenblatt solution remains finite.","tokens_in":681,"feed_emoji":"🎲","tokens_out":3316,"duration_ms":38312,"temperature":0.7,"pith_summary":"The paper is trying to show that the Leibenson equation — a single equation that generalizes both the porous-medium and p-Laplace equations — is a nonlinear Fokker-Planck equation, and that its self-similar Barenblatt source solutions are exactly the one-dimensional time-marginal densities of the unique strong solutions of a McKean-Vlasov stochastic differential equation. The SDE is unusual: its coefficients depend pointwise on the marginal density and on the density's first and second space derivatives, so the equation is meaningful only if those densities are regular enough. If the representation holds, the PDE's spreading and degenerate-diffusion behavior can be studied through trajectories of a single Markov-type process, which the authors call the Leibenson process. A second main claim is that this process is probabilistically strong: the solution is a measurable functional of the driving Brownian motion and initial condition, despite the degenerate diffusion and rough drift.","feed_headline":"Barenblatt solutions are the marginals of a single stochastic process","feed_subtitle":"A density-dependent McKean-Vlasov SDE encodes the nonlinear PDE, giving strong pathwise solutions.","key_machinery":"The key object is the McKean-Vlasov SDE whose coefficients are evaluated at the path's own time-marginal density and its first two spatial derivatives, making the stochastic equation a closure of the nonlinear Fokker-Planck picture. The Barenblatt profile — the explicit self-similar source solution of the PDE — is the benchmark this representation must reproduce, and the nonlinear Markov process built from these marginals is the 'Leibenson process' that carries the representation.","core_discovery":"The central claim is a probabilistic representation with two parts. First, the Leibenson equation $\\partial_t u = \\Delta_p u^q$ is identified as the nonlinear Fokker-Planck equation associated with a McKean-Vlasov SDE whose coefficients are functionals of the time-marginal density and its first and second derivatives. Second, the Barenblatt solutions of the PDE are represented as the one-dimensional marginal density curve $u(t,\\cdot)$ of those SDE solutions, and the resulting family of laws forms a nonlinear Markov process. The paper also asserts that these solutions are strong in the probabilistic sense — measurable with respect to the driving Brownian motion and initial condition — even th","pith_inferences":["One testable extension is that a particle system approximating the McKean-Vlasov SDE could produce numerical Barenblatt solutions in regimes where direct PDE solvers struggle with the degenerate interface.","The paper's coefficient structure — depending pointwise on derivatives of the marginal density — defines a new class of mean-field SDEs that may appear in other nonlinear diffusion problems beyond this equation.","The probabilistic representation may suggest a notion of weak solution for the Leibenson equation in low-regularity regimes, defined through SDE paths rather than PDE a priori estimates."],"forward_implications":["If the representation is correct, the Barenblatt solutions of the Leibenson equation can be obtained pathwise from a stochastic equation, not only from PDE analysis.","The existence of strong solutions means the PDE's probabilistic counterpart has a pathwise meaning, not just a law-level meaning, despite the irregular coefficients.","The results extend the stochastic approach that works for the porous-medium and p-Laplace equations to a single equation that contains both as special cases.","The well-posedness of the SDE is tied to regularity of the marginal densities, so the paper's PDE regularity analysis and stochastic existence proof support each other."],"supporting_citations":[],"fun_headline_variants":["Leibenson equation gets a stochastic process counterpart","Barenblatt solutions are density marginals of a Markov process","Novel McKean-Vlasov SDE yields strong solutions to degenerate PDE","Leibenson process: strong stochastic solutions to nonlinear PDEs"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The main load-bearing premise is that the time-marginal densities of the McKean-Vlasov solution are smooth enough for their first and second spatial derivatives to appear as pointwise coefficients in the SDE; if any density loses enough regularity, the SDE is not even well-posed.","fun_headline_variants_meta":{"raw":{"variants":["Leibenson equation gets a stochastic process counterpart","Barenblatt solutions are density marginals of a Markov process","Novel McKean-Vlasov SDE yields strong solutions to degenerate PDE","Leibenson process: strong stochastic solutions to nonlinear PDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000181,"raw_usage":{"total_tokens":1155,"prompt_tokens":768,"completion_tokens":387,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":324}},"tokens_in":512,"tokens_out":387,"duration_ms":4755,"temperature":1.0,"reasoning_tokens":324,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:08:50.484749+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a parameter regime of strong degeneracy where the Barenblatt solution has a non-smooth interface or cusp. If the first or second spatial derivative of the marginal density fails to be a well-defined pointwise function at the interface, the coefficient evaluation in the McKean-Vlasov SDE is undefined and the claimed representation cannot hold; one could look for a time where the simulated marginal's derivative diverges while the PDE's Barenblatt solution remains finite.","supporting_citations":[],"review_version":1}