Pith. sign in

REVIEW 1 major objections 4 minor 56 references

Nonlinear parabolic characterizations of stochastic completeness at infinity on weighted graphs

T0 review · 1 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read A weighted graph is stochastically complete at infinity exactly when every bounded initial datum for the filtration equation has a unique bounded pointwise solution, and failure of the property produces infinitely many such solutions.

desk verdict Genuinely new graph analogue of the nonlinear parabolic characterization of stochastic completeness; solid internal proofs, but the main theorem is conditional on the companion preprint [4] — worth refereeing. read the letter →

arxiv 2608.11931 v1 pith:63S5K2UX submitted 2026-08-12 math.PR math.AP

Davide Bianchi , Bobo Hua , Alberto G. Setti , Rados{l}aw K. Wojciechowski This is my paper · ORCID
classification math.PRmath.AP MSC 35K5535R0205C6360J27
keywords stochasticcompletenessatinfinityweightedgraphsgeneralizedporousmediumequationfiltrationuniquenessofboundedpointwisesolutionsmassbalancegraphLaplacianOmori–Yaumaximumprinciple
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 proves a nonlinear version of a classical probabilistic notion for weighted graphs: heat is lost only through the killing term and never at infinity exactly when a nonlinear filtration equation has at most one bounded solution for every bounded starting profile. The equation is $(\partial_t+\Delta\Phi)u=0$, where $\Delta$ is the graph Laplacian and $\Phi u=\varphi\circ u$ for a continuous, increasing, nonconstant function $\varphi$ with $\varphi(0)=0$; it includes the heat equation, the porous medium powers, and the fast diffusion powers. The main theorem says stochastic completeness at infinity holds if and only if every bounded initial datum admits a unique bounded pointwise solution on every finite time horizon, and when the property fails, every bounded datum admits infinitely many distinct bounded solutions. A second theorem detects the same property through a generalized mass balance: the mass remaining at time $t$ plus the mass removed by killing up to $t$ equals the initial mass. These results extend manifold parabolic characterizations to weighted graphs without assuming local finiteness.

What carries the argument

The load-bearing mechanism is the trapping of all bounded solutions between two extremal solutions. For constants $A$ and $B$ bracketing the initial datum, the companion construction [4] supplies global bounded pointwise solutions $u_A\le u_B$ such that every bounded solution lying between $A$ and $B$ satisfies $u_A\le v\le u_B$; uniqueness under stochastic completeness at infinity therefore reduces to proving $u_A=u_B$. Subtracting the equations gives $q=u_B-u_A$ and $\rho=\varphi(u_B)-\varphi(u_A)$ with $\partial_t q+\Delta \rho=0$, and a concave modulus of continuity $\omega$ for $\varphi$ on $[A,B]$ (Lemma 4.1) gives the pointwise bound $0\le \rho\le \omega(q)$. Lemma 4.2 turns any such pair into a bounded function $W$ with $-\Delta W\ge \omega^{-1}(W)$; positivity of $W$ would contradict the weak Omori–Yau maximum principle, which is equivalent to stochastic completeness at infinity. In the incomplete case a normalized defect function $V$, with $\sup_X V=1$ and $\Delta V\le -(V+1)-K$ for $K=\kappa/\mu$, builds barriers that separate, on the super-level sets of $V$, the limits obtained from finite-graph exhaustions with different constant exterior data, forcing infinitely many distinct solutions.

What would settle it

Take a stochastically complete at infinity birth–death chain (for instance the chain in Example 7.7(a)) and run the finite-graph exhaustion construction with two different constant exterior data for $\varphi(s)=s|s|^{-1/2}$ and a compactly supported bounded initial datum. The theorem predicts the two subsequential limits coincide as bounded pointwise solutions on every $[0,T]$; finding two different bounded limits on some positive time interval would falsify the uniqueness half, while uniqueness of the limit corroborates it.

Watch

Extended reading notes

Core claim

For any weighted graph $G=(X,w,\kappa,\mu)$, let $\Delta$ be the formal nonnegative Laplacian and consider the filtration equation $(\partial_t+\Delta\Phi)u=0$ with $\Phi u=\varphi\circ u$ for $\varphi$ in the class $\mathcal I$ of continuous increasing nonconstant functions vanishing at zero. The central result (Theorem 6.2) proves that $G$ satisfies stochastic completeness at infinity if and only if, for every $T>0$, every $u_0\in\ell^\infty(X)$, and every $\varphi\in\mathcal I$, the equation has a unique bounded pointwise solution on $[0,T]\times X$; detection is possible with a single datum, with the zero datum alone, and by a global comparison principle. If $G$ fails the property, Theorem 5.2 constructs, for every $\varphi\in\mathcal I$ and every bounded datum, infinitely many global bounded pointwise solutions that are pairwise distinct on every finite time interval. Theorem 7.4 adds an equivalent mass formulation: under finite total measure or under the sharp linear-growth condition $\limsup_{r\downarrow 0}\varphi(r)/r<\infty$, stochastic completeness at infinity is equivalent to the generalized mass balance (MB) holding for every bounded positive solution with finite-mass data; when $\kappa=0$ the balance becomes conservation of mass.

Load-bearing premise

The proof imports from the companion study [4] the guarantee that extremal bounded solutions exist, trap all other bounded solutions between them, and obey a finite-subgraph comparison principle for every nonlinearity, killing term, and graph in the stated generality; if any of those companion results fails, the characterization collapses.

Editorial extensions

If this is right

  • For the identity nonlinearity, the characterization reduces to the classical linear statement: stochastic completeness at infinity is equivalent to uniqueness of bounded heat-equation solutions, and when the killing term vanishes this is ordinary stochastic completeness.
  • Uniqueness for the linear heat equation automatically implies uniqueness for every nonlinearity in the allowed class, so nonlinear effects cannot create new ambiguity once heat is known not to escape.
  • On graphs that are stochastically incomplete at infinity, every bounded initial datum, including zero, has infinitely many bounded solutions on every finite time interval; for zero initial data the extra solutions can be chosen of one sign.
  • The mass-balance theorem ties the property to an observable conservation law: on finite-measure graphs or under linear growth of the nonlinearity at zero, the total mass of a bounded positive solution plus the mass removed by killing equals the initial mass for every finite-mass datum.

Reading between the lines

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

  • Beyond the paper: the single-datum and zero-datum detection in Theorem 6.2 suggests a computational test on any candidate graph: integrate the filtration equation from zero initial datum and look for two bounded profiles, which would certify stochastic incompleteness at infinity numerically.
  • Beyond the paper: Example 7.7 shows that on infinite-measure graphs the fast-diffusion range can break the mass balance even under stochastic completeness, so the linear-growth condition marks a real phase boundary where nonlinear flux outgrows summability before the property itself is lost.
  • Beyond the paper: the measure-change Proposition 7.8 converts any known criterion for ordinary stochastic completeness of killing-free graphs into a criterion for stochastic completeness at infinity by replacing the vertex measure with the vertex measure plus the killing weight, which may yield new geometric sufficient conditions.
  • Beyond the paper: the dichotomy suggests that non-uniqueness at infinity is a generic nonlinear signature of boundary leakage, so similar characterizations may hold for other Markov generators and for nonlinearities satisfying only mild continuity and monotonicity.
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

1 major / 4 minor

Summary. The paper studies the generalized porous medium (filtration) equation (∂t + ΔΦ)u = 0 on weighted graphs with a formal Laplacian that includes a killing term κ, where Φu = φ∘u for a continuous, increasing, nonconstant φ with φ(0)=0. The main claim (Theorem A / Theorem 6.2) is that stochastic completeness at infinity (SC∞) is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum and every admissible nonlinearity; if SC∞ fails, every bounded initial datum admits infinitely many global bounded pointwise solutions, pairwise distinct on every finite time interval (Theorem 5.2). The paper also proves a mass-balance characterization (Theorem B / Theorem 7.4): under finite total measure or the linear-growth condition (LG), SC∞ is equivalent to the generalized mass balance (MB) holding for every bounded positive solution with finite-mass datum, with sharpness examples showing that both the growth condition and boundedness are needed. A final measure-change result (Proposition 7.8) reduces SC∞ to ordinary stochastic completeness after absorbing the killing term into the vertex measure.

Significance. If correct, this resolves the nonlinear parabolic characterization problem for graphs raised by Schmidt–Zimmermann and extends the manifold results of Grillo–Ishige–Muratori–Punzo to weighted graphs with killing and no local-finiteness assumption. The internal proofs are detailed and careful: the modulus-of-continuity argument in Lemma 4.2, the barrier construction in Theorem 5.2, and the no-flux identity in Lemma 7.1 are all substantive and appear correct. The paper is also honest about its hypotheses and provides explicit birth–death chain examples showing that the growth and boundedness assumptions in the mass-balance theorem are sharp. There are no fitted parameters and no circular deductions; the equivalences are genuinely new statements. The principal weakness is that the paper is not self-contained: the finite-subgraph comparison principle and the existence/trapping results on which the central dichotomy rests are imported from the companion preprint [4].

major comments (1)
  1. [Section 3, Lemma 3.1 and Theorem 3.3] The central 'if and only if' is conditional on the companion preprint [4]. Lemma 3.1 (finite-subgraph parabolic comparison) and Theorem 3.3 (existence of global bounded pointwise solutions together with the trapping property (3.1)) are stated without proof. These are load-bearing in both directions of Theorem 6.2: Theorem 4.3 uses the trapping property (3.1) to reduce arbitrary bounded solutions v1,v2 to the extremal pair uA,uB, and Theorem 5.2 uses Lemma 3.1 together with the finite-subgraph construction from [4, proof of Theorem 3.4]. Theorem 7.4 also uses Lemma 3.1 repeatedly. Since [4] is an unreviewed companion preprint, the manuscript as submitted does not provide a complete proof of its main characterization. The authors should either include full proofs of Lemma 3.1 and Theorem 3.3 in an appendix, or state precisely which conditions on the graph (non-local finiteness, arbitrary κ≥0) are verified in [4] and make the paper's acceptance conditional on the companion being published in final form.
minor comments (4)
  1. [Section 3 / Section 7] The notation 'u_0' for the minimal positive solution in Theorem 3.3 clashes with the notation for the initial datum u0; this becomes confusing in the proof of Theorem 7.4, where 'u0 = lim_n U_n' is used. Please use a different symbol, e.g. u_min or u_*.
  2. [Section 5, Step 2] The inequalities in (5.7) are automatically satisfied because β_α ≤ β_α and β_α ≥ β_α; as written they are vacuous. Either remove them or state the intended nontrivial inequalities explicitly.
  3. [Theorem 7.4, proof of Step 2] The proof says 'a common diagonal extraction' produces ordered limit solutions u1 ≤ u2 for the two exterior values 0 and α2. This requires a simultaneous Arzelà–Ascoli extraction for the pair of sequences; the details should be spelled out, since the two finite-subgraph solutions are ordered for each n by Lemma 3.1 but the extraction must preserve the ordering in the limit.
  4. [Section 5, Step 1] In the finite ODE argument, Peano's theorem gives only local existence; the comparison principle is then used to show that solutions remain in a fixed bounded box and hence extend globally. This implication should be stated explicitly, because it is the comparison principle that supplies the global extension.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main equivalence is genuinely new and is built on independently stated well-posedness results from the companion preprint [4], whose assumptions do not include stochastic completeness at infinity.

full rationale

The paper's central result (Theorem 6.2) equates stochastic completeness at infinity with uniqueness of bounded pointwise solutions of the filtration equation. The uniqueness direction (Theorem 4.3) is proved internally: it reduces two arbitrary solutions to the extremal pair via the trapping property imported from [4, Theorem 3.4], then applies an original modulus-of-continuity argument (Lemmas 4.1 and 4.2) that uses the Omori–Yau characterization of SC_infinity from the published literature. The nonuniqueness direction (Theorem 5.2) is also proved in the paper via an exhaustion construction with constant exterior data and a barrier separation argument; the only imported ingredients are the finite-subgraph parabolic comparison principle (Lemma 3.1) and the existence/extremality of bounded solutions (Theorem 3.3), both stated as results of the companion preprint [4]. Although [4] is by overlapping authors, its stated hypotheses (arbitrary graph, phi in I, bounded initial datum) do not include stochastic completeness at infinity, so citing it is not a reduction of the target claim to itself. No parameter is fitted and no quantity called a prediction is defined in terms of the target. The paper is transparent that Section 3 summarizes [4], and all remaining steps are self-contained. Reliance on an unverified companion preprint is a completeness or correctness risk, not circularity, under the rule that a cited parameter-free result with assumptions not containing the target is independent support.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities: this is a structural theorem. The central claims rest on standard Dirichlet-space theory plus one companion-paper machinery; the true novelty is the equivalence proof itself.

assumptions (4)
  • domain assumption Standing graph hypotheses: X countable and connected, mu>0, symmetric edge weights w with sum_y w(x,y)<infinity for every x, killing kappa>=0.
    Defines the class of weighted graphs in Section 2; all theorems are conditional on it.
  • domain assumption Finite-subgraph comparison principle and existence/trapping of extremal bounded pointwise solutions from companion paper [4] (Lemma 3.1 and Theorem 3.3).
    Imported unproved results that Theorems 4.3 and 5.2 rely on; they are stated verbatim in Section 3.
  • standard math Known equivalences for SC_infinity (Proposition 2.4): beta-injectivity on l^infty, bounded-heat uniqueness, weak Omori-Yau principle; from [33], [34], and [28].
    Used in Lemma 4.2 for the Omori-Yau contradiction and in Lemma 7.1 for the resolvent identity.
  • standard math Monotone resolvent and semigroup extensions, plus the generalized conservation identity 1 = beta R_beta 1 + R_beta K from Keller-Lenz theory.
    Basis of Lemma 7.1 no-flux identity in the mass-balance half.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nonlinear parabolic characterizations of stochastic completeness at infinity on weighted graphs." pith.science (2026). https://pith.science/paper/63S5K2UX

@misc{pith2026260811931,
  author       = {Pith},
  title        = {Pith review of: Nonlinear parabolic characterizations of stochastic completeness at infinity on weighted graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/63S5K2UX}},
  note         = {Machine review of arXiv:2608.11931}
}
abstract

We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + \Delta \Phi)u =0 \] where $\Delta$ is the non-negative formal graph Laplacian and $\Phi u =\phi \circ u$ with $\phi \colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For \(\Phi=\id\), this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when \(\kappa=0\). If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $\phi(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}\kappa(x)\phi(u(s,x)) \dd s$ dissipated by the killing term $\kappa$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}\phi(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 26 canonical work pages

  1. [4]

    The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation

    D. Bianchi, B. Hua, A. G. Setti, and R. K. Wojciechowski.The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation. 2026. arXiv:2607.23091 [math.AP]

    work page Pith review arXiv 2026

  2. [1]

    The L1-Liouville property on graphs

    A. Adriani and A. G. Setti. “The L1-Liouville property on graphs”. In:Journal of Fourier Analysis and Applications29.4 (2023), Paper No. 44, 20 pp.DOI:10.1007/s00041-023-10025-3

    work page doi:10.1007/s00041-023-10025-3 2023

  3. [2]

    Semilinear diffusion equations on infinite graphs: the dissipative and Lipschitz cases

    E. Berchio, D. Bianchi, A. G. Setti, and M. Vallarino. “Semilinear diffusion equations on infinite graphs: the dissipative and Lipschitz cases”. In:Journal of Differential Equations476.Part 2 (2026), Paper No. 114485.DOI: 10.1016/j.jde.2026.114485

    work page arXiv 2026

  4. [3]

    Berchio, F

    E. Berchio, F. Santagati, and M. Vallarino.The fractional Porous Medium Equation on graphs. 2026. arXiv: 2606.23360 [math.AP]

    arXiv 2026

  5. [5]

    Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

    D. Bianchi and A. G. Setti. “Laplacian cut-offs, porous and fast diffusion on manifolds and other applications”. In:Calculus of Variations and Partial Differential Equations57.1 (2018), Paper No. 4, 33 pp.DOI: 10.1007/ s00526-017-1267-9

    work page 2018

  6. [6]

    The generalized porous medium equation on graphs: existence and uniqueness of solutions withℓ1 data

    D. Bianchi, A. G. Setti, and R. K. Wojciechowski. “The generalized porous medium equation on graphs: existence and uniqueness of solutions withℓ1 data”. In:Calculus of Variations and Partial Differential Equations61.5 (2022), Paper No. 171, 42 pp.DOI:10.1007/s00526-022-02249-w

    doi:10.1007/s00526-022-02249-w 2022

  7. [7]

    TheL∞-positivity preserving property and stochastic completeness

    A. Bisterzo and L. Marini. “TheL∞-positivity preserving property and stochastic completeness”. In:Potential Analysis59.4 (2023), pp. 2017–2034.DOI:10.1007/s11118-022-10041-w

    work page doi:10.1007/s11118-022-10041-w 2023

  8. [8]

    Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

    M. Bonforte and J. L. V´azquez. “Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations”. In:Advances in Mathematics223.2 (2010), pp. 529–578.DOI:10.1016/j.aim.2009.08.021

    doi:10.1016/j.aim.2009.08.021 2010

Show all 56 references
  1. [9]

    Heat kernel bounds, conservation of probability and the Feller property

    E. B. Davies. “Heat kernel bounds, conservation of probability and the Feller property”. In:Journal d’Analyse Math´ematique58.1 (1992), pp. 99–119.DOI: 10.1007/BF02790359. Festschrift on the occasion of the 70th birthday of Shmuel Agmon

    1992 doi

  2. [10]

    V olume growth and stochastic completeness of graphs

    M. Folz. “V olume growth and stochastic completeness of graphs”. In:Transactions of the American Mathematical Society366.4 (2014), pp. 2089–2119.DOI:10.1090/S0002-9947-2013-05930-2

    2014 doi

  3. [11]

    Aronson-B´enilan gradient estimates for porous medium equations under lower bounds of N- weighted Ricci curvature with N <0

    Y . Fujitani. “Aronson-B´enilan gradient estimates for porous medium equations under lower bounds of N- weighted Ricci curvature with N <0 ”. In:Kyushu Journal of Mathematics78.1 (2024), pp. 209–223.DOI: 10.2206/kyushujm.78.209. 24 Bianchi · Hua · Setti · Wojciechowski Nonline...

    2024 doi

  4. [12]

    The conservation property of the heat equation on Riemannian manifolds

    M. P. Gaffney. “The conservation property of the heat equation on Riemannian manifolds”. In:Communications on Pure and Applied Mathematics12.1 (1959), pp. 1–11.DOI:10.1002/cpa.3160120102

    1959 doi

  5. [13]

    Stochastic completeness andL1-Liouville property for second- order elliptic operators

    D. Ganguly, Y . Pinchover, and P. Roychowdhury. “Stochastic completeness andL1-Liouville property for second- order elliptic operators”. In:Discrete and Continuous Dynamical Systems - S17.5 & 6 (2024), pp. 1672–1685. DOI:10.3934/dcdss.2022138

    2024 doi

  6. [14]

    Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

    A. Grigor’yan. “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds”. In:Bulletin of the American Mathematical Society36.2 (1999), pp. 135–249.DOI: 10.1090/S0273-0979-99-00776-4

    1999 doi

  7. [15]

    Grigor’yan.Heat Kernel and Analysis on Manifolds

    A. Grigor’yan.Heat Kernel and Analysis on Manifolds. V ol. 47. AMS/IP Studies in Advanced Mathematics. Providence, RI and Boston, MA: American Mathematical Society and International Press, 2009. 482 pp.ISBN: 978-0-8218-9393-7.DOI:10.1090/amsip/047

    2009 doi

  8. [16]

    Stochastically complete manifolds

    A. Grigor’yan. “Stochastically complete manifolds”. In:Doklady Akademii Nauk SSSR290.3 (1986), pp. 534–

    1986

  9. [17]

    Kazdan–Warner equation on graph

    A. Grigor’yan, Y . Lin, and Y . Yang. “Kazdan–Warner equation on graph”. In:Calculus of Variations and Partial Differential Equations55.4 (2016), Paper No. 92, 13 pp.DOI:10.1007/s00526-016-1042-3

    2016 doi

  10. [18]

    Yamabe type equations on graphs

    A. Grigor’yan, Y . Lin, and Y . Yang. “Yamabe type equations on graphs”. In:Journal of Differential Equations 261.9 (2016), pp. 4924–4943.DOI:10.1016/j.jde.2016.07.011

    2016 doi

  11. [19]

    Nonlinear characterizations of stochastic completeness

    G. Grillo, K. Ishige, and M. Muratori. “Nonlinear characterizations of stochastic completeness”. In:Journal de Math´ematiques Pures et Appliqu´ees139 (2020), pp. 63–82.DOI:10.1016/j.matpur.2020.05.008

    2020 doi

  12. [20]

    A general nonlinear characterization of stochastic incomplete- ness

    G. Grillo, K. Ishige, M. Muratori, and F. Punzo. “A general nonlinear characterization of stochastic incomplete- ness”. In:Journal de Math ´ematiques Pures et Appliqu´ees206 (2026), Paper No. 103839.DOI: 10.1016/j. matpur.2025.103839

    2026

  13. [21]

    Blow-up and global existence for semilinear parabolic equations on infinite graphs

    G. Grillo, G. Meglioli, and F. Punzo. “Blow-up and global existence for semilinear parabolic equations on infinite graphs”. In:Calculus of Variations and Partial Differential Equations65.4 (2026), Paper No. 114, 20 pp.DOI: 10.1007/s00526-026-03291-8

    2026 doi

  14. [22]

    The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach

    G. Grillo, D. D. Monticelli, and F. Punzo. “The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach”. In:Journal of Differential Equations430 (2025), Paper No. 113191.DOI:10.1016/j.jde.2025.02.062

    2025 doi

  15. [23]

    Smoothing effects for the porous medium equation on Cartan–Hadamard manifolds

    G. Grillo and M. Muratori. “Smoothing effects for the porous medium equation on Cartan–Hadamard manifolds”. In:Nonlinear Analysis131 (2016), pp. 346–362.DOI:10.1016/j.na.2015.07.029

    2016 doi

  16. [24]

    A porous medium equation with rough weights: sharp Widder theory

    G. Grillo, M. Muratori, T. Petitt, and N. Simonov. “A porous medium equation with rough weights: sharp Widder theory”. In:Transactions of the American Mathematical Society(2026).DOI:10.1090/tran/9787

    2026 doi

  17. [25]

    Global properties of Dirichlet forms in terms of Green’s formula

    S. Haeseler, M. Keller, D. Lenz, J. Masamune, and M. Schmidt. “Global properties of Dirichlet forms in terms of Green’s formula”. In:Calculus of Variations and Partial Differential Equations56.5 (2017), Paper No. 124, 43 pp.DOI:10.1007/s00526-017-1216-7

    2017 doi

  18. [26]

    Stochastic completeness for graphs with curvature dimension conditions

    B. Hua and Y . Lin. “Stochastic completeness for graphs with curvature dimension conditions”. In:Advances in Mathematics306 (2017), pp. 279–302.DOI:10.1016/j.aim.2016.10.022

    2017 doi

  19. [27]

    Time regularity and long-time behavior of parabolicp-Laplace equations on infinite graphs

    B. Hua and D. Mugnolo. “Time regularity and long-time behavior of parabolicp-Laplace equations on infinite graphs”. In:Journal of Differential Equations259.11 (2015), pp. 6162–6190.DOI: 10.1016/j.jde.2015.07. 018

    2015 doi

  20. [28]

    Stochastic incompleteness for graphs and weak Omori–Yau maximum principle

    X. Huang. “Stochastic incompleteness for graphs and weak Omori–Yau maximum principle”. In:Journal of Mathematical Analysis and Applications379.2 (2011), pp. 764–782.DOI:10.1016/j.jmaa.2011.02.009

    2011 doi

  21. [29]

    On the uniqueness class, stochastic completeness and volume growth for graphs

    X. Huang, M. Keller, and M. Schmidt. “On the uniqueness class, stochastic completeness and volume growth for graphs”. In:Transactions of the American Mathematical Society373.12 (2020), pp. 8861–8884.DOI: 10.1090/tran/8211

    2020 doi

  22. [30]

    Extinction profiles for the Sobolev critical fast diffusion equation in bounded domains. I. One bubble dynamics

    T. Jin and J. Xiong. “Extinction profiles for the Sobolev critical fast diffusion equation in bounded domains. I. One bubble dynamics”. In:Archive for Rational Mechanics and Analysis250.2 (2026), Paper No. 14.DOI: 10.1007/s00205-026-02177-5

    2026 doi

  23. [31]

    Singular extinction profiles of solutions to some fast diffusion equations

    T. Jin and J. Xiong. “Singular extinction profiles of solutions to some fast diffusion equations”. In:Journal of Functional Analysis283.7 (2022), Paper No. 109595.DOI:10.1016/j.jfa.2022.109595

    2022

  24. [32]

    The heat equation on complete Riemannian manifolds

    L. Karp and P. Li. “The heat equation on complete Riemannian manifolds”. Unpublished manuscript. 1983

    1983

  25. [33]

    Dirichlet forms and stochastic completeness of graphs and subgraphs

    M. Keller and D. Lenz. “Dirichlet forms and stochastic completeness of graphs and subgraphs”. In:Journal f¨ur die reine und angewandte Mathematik (Crelles Journal)666 (2012), pp. 189–223.DOI: 10.1515/CRELLE. 2011.122. 25 Bianchi · Hua · Setti · Wojciechowski Nonlinear paraboli...

    2012 doi

  26. [34]

    Keller, D

    M. Keller, D. Lenz, and R. K. Wojciechowski.Graphs and Discrete Dirichlet Spaces. V ol. 358. Grundlehren der mathematischen Wissenschaften. Cham: Springer, 2021. 668 pp.ISBN: 978-3-030-81459-5.DOI: 10.1007/978- 3-030-81459-5

    2021 doi

  27. [35]

    Neumann semigroup, subgraph convergence, form uniqueness, stochastic completeness and the Feller property

    M. Keller, F. M¨unch, and R. K. Wojciechowski. “Neumann semigroup, subgraph convergence, form uniqueness, stochastic completeness and the Feller property”. In:The Journal of Geometric Analysis35.1 (2025), Paper No. 14, 23 pp.DOI:10.1007/s12220-024-01838-9

    2025 doi

  28. [36]

    Porous medium equations with reactions on a metric graph

    R. Kou, P. Lai, and B. Lou. “Porous medium equations with reactions on a metric graph”. In:Discrete and Continuous Dynamical Systems - B38 (2026), pp. 448–462.DOI:10.3934/dcdsb.2026079

    2026 doi

  29. [37]

    New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds

    K. Kuwae and X.-D. Li. “New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds”. In:Bulletin of the London Mathematical Society54.2 (2022), pp. 404–427.DOI: 10.1112/blms.12568

    2022 doi

  30. [38]

    Blow-up of nonnegative solutions of an abstract semilinear heat equation with convex source

    D. Lenz, M. Schmidt, and I. Zimmermann. “Blow-up of nonnegative solutions of an abstract semilinear heat equation with convex source”. In:Calculus of Variations and Partial Differential Equations62.4 (2023), Paper No. 140, 19 pp.DOI:10.1007/s00526-023-02482-x

    2023 doi

  31. [39]

    The existence and nonexistence of global solutions for a semilinear heat equation on graphs

    Y . Lin and Y . Wu. “The existence and nonexistence of global solutions for a semilinear heat equation on graphs”. In:Calculus of Variations and Partial Differential Equations56.4 (2017), Paper No. 102, 22 pp.DOI: 10.1007/s00526-017-1204-y

    2017 doi

  32. [40]

    Porous media equation on locally finite graphs

    L. Ma. “Porous media equation on locally finite graphs”. In:Archivum Mathematicum58.3 (2022), pp. 177–187. DOI:10.5817/AM2022-3-177

    2022 doi

  33. [41]

    A generalized conservation property for the heat semigroup on weighted manifolds

    J. Masamune and M. Schmidt. “A generalized conservation property for the heat semigroup on weighted manifolds”. In:Mathematische Annalen377.3-4 (2020), pp. 1673–1710.DOI: 10.1007/s00208-019-01888- 3

    2020 doi

  34. [42]

    Evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

    J. M. Maz ´on, M. Solera, and J. Toledo. “Evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces”. In:Nonlinear Analysis197 (2020), Paper No. 111813.DOI:10.1016/j.na.2020.111813

    2020

  35. [43]

    Nonexistence of solutions to parabolic problems with a potential on weighted graphs

    D. D. Monticelli, F. Punzo, and J. Somaglia. “Nonexistence of solutions to parabolic problems with a potential on weighted graphs”. In:Journal of Differential Equations453.Part 1 (2026), Paper No. 113782.DOI: 10.1016/ j.jde.2025.113782

    2026

  36. [44]

    Parabolic theory of the discretep-Laplace operator

    D. Mugnolo. “Parabolic theory of the discretep-Laplace operator”. In:Nonlinear Analysis: Theory, Methods & Applications87 (2013), pp. 33–60.DOI:10.1016/j.na.2013.04.002

    2013 doi

  37. [45]

    Ollivier Ricci curvature for general graph Laplacians: heat equation, Laplacian comparison, non-explosion and diameter bounds

    F. M¨unch and R. K. Wojciechowski. “Ollivier Ricci curvature for general graph Laplacians: heat equation, Laplacian comparison, non-explosion and diameter bounds”. In:Advances in Mathematics356 (2019), Paper No. 106759, 45 pp.DOI:10.1016/j.aim.2019.106759

    2019

  38. [46]

    A remark on the maximum principle and stochastic completeness

    S. Pigola, M. Rigoli, and A. G. Setti. “A remark on the maximum principle and stochastic completeness”. In: Proceedings of the American Mathematical Society131.4 (2003), pp. 1283–1288.DOI: 10.1090/S0002-9939- 02-06672-8

    2003 doi

  39. [47]

    Pigola, M

    S. Pigola, M. Rigoli, and A. G. Setti.Maximum Principles on Riemannian Manifolds and Applications. V ol. 174. Memoirs of the American Mathematical Society 822. Providence, RI: American Mathematical Society, 2005. 99 pp.ISBN: 978-0-8218-3639-2.DOI:10.1090/memo/0822

    2005 doi

  40. [48]

    On a semilinear parabolic equation with time-dependent source term on infinite graphs

    F. Punzo and A. Sacco. “On a semilinear parabolic equation with time-dependent source term on infinite graphs”. In:Journal of Evolution Equations26.1 (2026), Paper No. 13, 16 pp.DOI:10.1007/s00028-025-01159-6

    2026 doi

  41. [49]

    A nonlinear characterization of stochastic completeness of graphs

    M. Schmidt and I. Zimmermann. “A nonlinear characterization of stochastic completeness of graphs”. In: Mathematische Nachrichten298.3 (2025), pp. 925–943.DOI:10.1002/mana.202400436

    2025 doi

  42. [50]

    J. L. V´azquez.The Porous Medium Equation: Mathematical Theory. Oxford Mathematical Monographs. Oxford: Clarendon Press, Oxford University Press, 2007.ISBN: 978-0-19-856903-9.DOI: 10.1093/acprof:oso/ 9780198569039.001.0001

    2007 doi

  43. [51]

    Some gradient estimates and Liouville properties of the fast diffusion equation on Riemannian manifolds

    W. Wang, R. Xie, and P. Zhang. “Some gradient estimates and Liouville properties of the fast diffusion equation on Riemannian manifolds”. In:Chinese Annals of Mathematics, Series B42.4 (2021), pp. 529–550.DOI: 10.1007/s11401-021-0276-3

    2021 doi

  44. [52]

    Analysis of the physical Laplacian and the heat flow on a locally finite graph

    A. Weber. “Analysis of the physical Laplacian and the heat flow on a locally finite graph”. In:Journal of Mathematical Analysis and Applications370.1 (2010), pp. 146–158.DOI:10.1016/j.jmaa.2010.04.044

    2010 doi

  45. [53]

    Stochastic completeness of graphs: bounded Laplacians, intrinsic metrics, volume growth and curvature

    R. K. Wojciechowski. “Stochastic completeness of graphs: bounded Laplacians, intrinsic metrics, volume growth and curvature”. In:Journal of Fourier Analysis and Applications27.2 (2021), Paper No. 30.DOI: 10.1007/s00041-021-09821-6

    2021 doi

  46. [54]

    Stochastic completeness of graphs

    R. K. Wojciechowski. “Stochastic completeness of graphs”. Ph.D. thesis. City University of New York, 2008. 87 pp.ISBN: 978-0549-58579-4. ProQuest LLC, Ann Arbor, MI. 26 Bianchi · Hua · Setti · Wojciechowski Nonlinear parabolic characterizations of stochastic completeness at in...

    2008

  47. [55]

    Blow-up for a semilinear heat equation with Fujita’s critical exponent on locally finite graphs

    Y . Wu. “Blow-up for a semilinear heat equation with Fujita’s critical exponent on locally finite graphs”. In: Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas115.3 (2021), Paper No. 133, 16 pp.DOI:10.1007/s13398-021-01075-7. 27

    2021 doi

  48. [537]

    English translation: Soviet Math. Dokl. 34 (1987), no. 2, pp. 310–313

    1987

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.