The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms

Ian Zimmermann; Marcel Schmidt

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The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms

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Pith's one-line read This paper proves that nonlinear Dirichlet forms admit an extended Dirichlet space under mild conditions, and that subcriticality and criticality are characterized, respectively, by a weighted Hardy inequality and by the vanishing of the…

desk verdict A well-built, honest extension of criticality theory to nonlinear Dirichlet forms; the central theorems are proved under an explicit and reasonable Delta-2 condition. read the letter →

arxiv 2501.18391 v2 pith:Q5MHJ3D7 submitted 2025-01-30 math.FA

Marcel Schmidt , Ian Zimmermann This is my paper · ORCID

classification math.FA MSC 31C2535J9246E3047H05

keywords nonlinearDirichletformextendedspaceLuxemburgseminormmodularcriticalitysubcriticalityHardyinequalityvariableexponentp-Laplacian

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

Dirichlet forms are the energy functionals behind diffusion, and in the linear theory the extended Dirichlet space is the natural home for potential theory and for the dichotomy between recurrence and transience. The paper proves that this tool exists for nonlinear Dirichlet forms — convex, order-preserving energy functionals whose resolvents are order preserving and $\mathrm{L}^\infty$-contractive — under mild conditions, and then uses it to characterize their global behavior. Subcriticality (transience) is shown to be equivalent to a weighted Hardy inequality and to completeness of the extended Dirichlet space. Criticality (recurrence) is reduced to the single condition that the constant function $1$ lies in the extended space with zero extended energy. These equivalences hold for the large class of forms satisfying the weak $\Delta_2$-condition, which includes the $p$-Laplacian and variable-exponent models.

What carries the argument

The machinery centres on the extended Dirichlet space: starting from the energy $E$, one takes its lower semicontinuous relaxation $E_e$ with respect to local convergence in measure, and on the modular space $M(E_e)=\mathrm{lin}\,D(E_e)$ one puts the Luxemburg seminorm $\|f\|_{L_e}=\inf\{\lambda>0 : E_e(f/\lambda)\le 1\}$. The generalized semimodular formalism gives the lower semicontinuity and completeness results that make this space behave like its linear predecessor. The Green operator $Gf=\lim_{\alpha\to0+}G_\alpha f$ and the weighted forms $E_w=E+\tfrac12\int |f|^2 w$ connect the space to Hardy inequalities. The weak $\Delta_2$-condition — $E(f_n)\to0$ implies $E(2f_n)\to0$ — is the switch that makes $E_e$-convergence agree with Luxemburg-norm convergence, which is what lets the proofs identify criticality with $\|1\|_{L_e}=0$.

What would settle it

Construct a nonlinear Dirichlet form satisfying the standing assumptions but not the weak $\Delta_2$-condition — for example an energy built from a convex growth function $\varphi$ with $\varphi(2t)/\varphi(t)$ unbounded — and check whether it can have $1\in M(E_e)$ with $\|1\|_{L_e}=0$ while some nonzero $f\in L^1_+$ satisfies $Gf<\infty$ on a set of positive measure. That combination would violate Theorems 4.16 and 4.17 and Corollary 4.18.

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Extended reading notes

Core claim

Let $E$ be a nonlinear Dirichlet form satisfying the weak $\Delta_2$-condition, and let $E_e$ be its lower semicontinuous relaxation to functions that are only measurable. The paper's central claim is that $E$ is subcritical exactly when the Green operator is finite for some strictly positive function, exactly when a weighted Hardy inequality $\int |f| w\,d\mu \le \|f\|_{L_e}$ holds, and exactly when the extended Dirichlet space $(M(E_e), \|\cdot\|_{L_e})$ is complete as a normed space. $E$ is critical exactly when $1 \in M(E_e)$ and $\|1\|_{L_e}=0$. If $E$ is irreducible, the two cases are exhaustive: there is no intermediate behavior. The existence of $E_e$ itself is established under either completeness of the modular space or the nonlinear Dirichlet form assumption, via a theory of generalized semimodulars and their Luxemburg seminorms.

Load-bearing premise

The load-bearing premise is the weak $\Delta_2$-condition: if energies $E(f_n)$ tend to zero, then the energies $E(2f_n)$ must also tend to zero. It is what lets the authors pass from zero extended energy to zero Luxemburg norm; without it, the criticality and subcriticality characterizations could fail.

Editorial extensions

If this is right

For any reflexive nonlinear Dirichlet form satisfying the weak $\Delta_2$-condition, subcriticality can be tested either by a weighted Hardy inequality or by completeness of the extended Dirichlet space; this gives concrete transience criteria for $p$-Laplacian energies and variable-exponent Sobolev energies.
Criticality becomes a one-point computation: the form is recurrent precisely when the constant $1$ has zero extended energy, generalizing the ground-state alternative known for $p$-Laplacians.
Under irreducibility the dichotomy of Corollary 4.18 excludes mixed behavior: every such form is either critical or subcritical.
The existence of equilibrium potentials, Choquet capacities, and the nest characterization of exceptional sets in Section 5 follows once subcriticality and reflexivity are assumed, giving a potential theory for nonlinear energies.

Reading between the lines

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

The same machinery could yield a practical transience test for variable-exponent $p(\cdot)$-Laplacians: compute or bound the optimal constant in the weighted Hardy inequality of Theorem 4.16; finiteness would certify subcriticality. The paper does not perform such computations.
Because the paper notes that a maximal ergodic inequality is missing in the nonlinear setting, the natural next step is to seek such an inequality; if it exists, subcriticality might be strengthened to finiteness of $Gf$ for every $f\in L^1_+$, matching the linear case.
The obstruction to $h$-transforms for nonlinear order-preserving forms suggests that the criticality theory for the wider class will need a different tool than excessive-function transforms; one could try shifting the analysis to the level of the Luxemburg seminorm, where the paper observes the needed contraction property can fail.

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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

Bottom line: this is a solid paper that delivers what it promises. The genuinely new thing is the construction of an extended Dirichlet space for nonlinear Dirichlet forms in the Cipriani-Grillo sense via lower semicontinuous relaxation on L0, plus the criticality/subcriticality characterization through the Luxemburg norm of the extended functional. The equivalence in Theorems 4.16 and 4.17 between subcriticality/criticality and Hardy-inequality/completeness conditions is real, and the authors are upfront that the weak Delta-2 condition is load-bearing. I checked the key steps: the transfer of weak Delta-2 to the extended functional is justified in Lemma A.20 and the appendix, and the stress-test note about the minor citation gap is fair—the needed transfer follows from Proposition A.17 and lower semicontinuity, so the argument holds.

The paper is also honest about its limitations. Remark 3.7(c) says plainly that they could not extend the existence result to general nonlinear order preserving forms. Remark 5.9(b) flags the restrictive condition on excessive functions. Section 6 defers some example details to prior work, which is a minor confidence knock but not a correctness issue. The variable-exponent and Cheeger-type examples are genuine and show the scope.

Where are the soft spots? The weak Delta-2 condition is the main one. It is explicit, it holds for p-homogeneous forms and the variable-exponent cases, and the paper does not pretend otherwise. But it is not automatic, so the criticality dichotomy in Corollary 4.18 is conditional. That is a real boundary of the theory, not a flaw in the proof. The second soft spot is the absence of a nonlinear Hopf maximal ergodic inequality; Remark 4.20 names this directly. That means the criticality characterization is not as clean as the bilinear one—subcriticality gives finiteness only for one strictly positive function, not all L1 functions. The authors say so.

The appendix on generalized semimodulars is useful and mostly self-contained. Reflexivity appears in places and is a genuine assumption, but the examples cover it.

For whom: anyone working on nonlinear potential theory, p-Laplacian-type energies, or variable-exponent Sobolev spaces. The paper deserves a serious referee. I would send it to review, and I would cite it if I were writing in this area. The proofs are long and not machine-checked, but honestly presented, and the central claims are proved in the text.

Referee Report

1 major / 3 minor

Summary. The paper develops a potential theory for nonlinear Dirichlet forms in the sense of Cipriani and Grillo. The main contributions are: (i) Theorem 3.5, which proves the existence of the extended Dirichlet space under either completeness of the modular space or the nonlinear Dirichlet form property; (ii) Theorem 4.16, which characterizes subcriticality via weighted Hardy inequalities and completeness of the extended modular space under a weak Delta-2 condition; (iii) Theorem 4.17, which characterizes criticality via 1 belonging to the extended modular space with zero Luxemburg norm; (iv) Corollary 4.18, the dichotomy between criticality and subcriticality for irreducible forms; and (v) Theorem 5.8 and related results on equilibrium potentials, capacities, and exceptional sets. The paper includes examples for variable exponent p(.)-Laplacians on Riemannian manifolds and for p-energies induced by strongly local regular Dirichlet forms. A substantial appendix develops the theory of generalized semimodulars and Luxemburg seminorms that is used throughout the main text.

Significance. If the results are correct, this paper provides a genuinely nonlinear extension of the classical criticality and potential theory for bilinear Dirichlet forms, replacing the vector-space structure of the extended Dirichlet space with the modular space and Luxemburg norm. The characterizations of criticality (||1||_{Le}=0) and subcriticality (Hardy inequality plus completeness) are clean, falsifiable statements with clear analogues in the linear theory. The paper is careful to state the exact assumptions (reflexivity, completeness, weak Delta-2) and is honest about limitations, for example in Remark 3.7(c) and Remark 5.9(b). The appendix on generalized semimodulars is a useful contribution in its own right. The examples show that the theory covers important classes of nonlinear energies, and the paper explicitly relates its capacities to those of Beznea--Beznea--Roeckner and Kuwae. The main proofs are detailed and, apart from the gap identified below, appear internally consistent. The paper is likely to be influential in the ongoing development of nonlinear Dirichlet form theory.

major comments (1)

[Section 3.2, proof of Theorem 3.5(A)] The proof of part (A) invokes Theorem A.26 and Remark A.27 to conclude that the embedding (M(E), ||·||_{L,r}) → L2(μ) is continuous. However, Theorem A.26 yields this continuity only as assertion (ii), and the route (i) & (iii) ⇒ (ii) requires the lower semicontinuity of the generalized semimodular, which is precisely the statement being proved. As written, the argument is circular. A direct proof is needed, for instance via the recession cone of the sublevel set: if the embedding were discontinuous, one could find f_n with ||f_n||_L → 0 but ||f_n||_2 = 1; then g_n = f_n/||f_n||_L would satisfy E(g_n) bounded and the unbounded closed convex set {E ≤ C} would contain a non-zero recession direction lying in ker ||·||_L, contradicting that ||·||_L is a norm on M(E). This gap affects both the existence of Ee and the claim E = Ee under condition (A), so it must be repaired.

minor comments (3)

[Appendix A, Lemma A.20] Lemma A.20 is stated for the lower semicontinuous relaxation on the same space, whereas Ee is the relaxation on L0(μ). The transfer to the L0-envelope is not spelled out; it follows from Proposition A.17 by taking approximating sequences, but the reader would benefit from an explicit statement or a one-line justification.
[Theorem 5.8, proof of (ii) ⇒ (i)] In the proof, the phrase 'Let h̃ ∈ D(Ee) with h̃ > 0 a.s.' should specify that h̃ is E-excessive, otherwise the assumption (ii) does not apply. The subsequent argument works with the E-excessive function h provided by the theorem, so this is only a wording issue.
[Theorem 4.17, proof of (ii) ⇒ (i)] The argument applies Proposition 4.2 with f = 1, but that proposition is stated for f ∈ M(E). Since 1 ∈ M(Ee) and ||1||_{Le}=0, the inequality ∫ |1| w dμ ≤ (1+K(w))||1||_{Le} must be obtained by approximation from M(E); the paper does not make this approximation step explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the criticality and subcriticality theorems are proved from explicit assumptions and self-contained appendix results.

full rationale

The central claims — existence of the extended nonlinear Dirichlet form (Theorem 3.5), the subcriticality characterization via Hardy inequalities and completeness (Theorem 4.16), and the criticality characterization via 1 belonging to the extended modular space with zero Luxemburg norm (Theorem 4.17) — are proved in the text from the standing assumptions plus the explicitly stated weak Delta-2 condition. The weak Delta-2 hypothesis is not imported from the conclusions; its extension to the extended functional is established directly in Lemma A.20 and Theorem A.23, and the specific reduction used later (Ee(G^{w_n}w_n) -> 0 implies ||G^{w_n}w_n||_{Le} -> 0) is justified by that appendix material. The Green-operator/Hardy-inequality equivalence in Proposition 4.2 is derived, not assumed, and Theorem 4.8 is proved using the new Theorem A.26, whose proof is included. The self-citations to the authors' earlier work (e.g., [36] for a simplified existence proof and [34] for prior quadratic-form versions of Theorem A.13/A.26) are provenance remarks; the mathematical content is restated or reproved in the paper. No uniqueness theorem, fitted input, or ansatz is imported from a self-citation to force the main results. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No numeric parameters are fitted; the results are conditional on the structural assumptions listed above, all of which are stated in the paper.

assumptions (8)

domain assumption E is symmetric with E(0)=0
Standing assumption after Section 2; Remark 3.1 explains how asymmetric forms are reduced.
domain assumption E is a proper lower semicontinuous convex functional on L2 satisfying the relevant Beurling-Deny criterion
Definition 2.1 of nonlinear order preserving and nonlinear Dirichlet forms.
domain assumption Reflexivity of the modular space (M(E), ||·||_L)
Needed in Lemma 3.3, Proposition 3.8, Theorems 4.8/4.9, and the minimizer arguments of Section 5; the paper states all results hold for reflexive forms.
domain assumption Weak Delta-2 condition for E
Assumed in Theorems 4.16 and 4.17 to equate E-convergence with Luxemburg-seminorm convergence (Lemma A.8) and used in the W>0 construction.
domain assumption Irreducibility of E
Assumed in Corollary 4.18 for the criticality/subcriticality dichotomy; Theorem 3.15 shows irreducibility controls ker ||·||_{Le}.
domain assumption X is a Hausdorff topological space and mu is a sigma-finite Borel measure
Standing assumptions in Section 5 for capacities and equilibrium potentials.
domain assumption Existence of an E-excessive h in D(Ee) with Nh excessive for all N
Used in Theorem 5.8 to identify capacity-zero sets with exceptional sets; Remark 5.9(b) concedes this is restrictive.
standard math Standard convex analysis and functional analysis facts
Banach-Saks, Mazur's lemma, Fenchel-Moreau, closed graph theorem, and duality mapping properties are used with standard references.

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Pith. "Pith review of The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms." pith.science (2026). https://pith.science/paper/Q5MHJ3D7

@misc{pith2026250118391,
  author       = {Pith},
  title        = {Pith review of: The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5MHJ3D7}},
  note         = {Machine review of arXiv:2501.18391}
}

read the original abstract

In this paper we establish the existence of the extended Dirichlet space for nonlinear Dirichlet forms under mild conditions. We employ it to introduce and characterize criticality (recurrence) and subcriticality (transience) and establish basics of a potential theory.

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Works this paper leans on

40 extracted references · 37 canonical work pages · cited by 1 Pith paper

[1]

Necessary and suﬃcient conditions for the exis- tence of m-perfect processes associated with Dirichlet forms

Sergio Albeverio and Zhi Ming Ma. Necessary and suﬃcient conditions for the exis- tence of m-perfect processes associated with Dirichlet forms. In S´ eminaire de Proba- bilit´ es, XXV, volume 1485 of Lecture Notes in Math. , pages 374–406. Springer, Berlin, 1991

work page 1991
[2]

A general corresponden ce between Dirichlet forms and right processes

Sergio Albeverio and Zhi Ming Ma. A general corresponden ce between Dirichlet forms and right processes. Bull. Amer. Math. Soc. (N.S.) , 26(2):245–252, 1992

work page 1992
[3]

Cal culus and heat ﬂow in metric measure spaces and applications to spaces with Ricci bounds from below

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´ e. Cal culus and heat ﬂow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math., 195(2):289–391, 2014

work page 2014
[4]

p-brownian motion and the p-laplacian

Viorel Barbu, Marco Rehmeier, and Michael R¨ ockner. p-brownian motion and the p-laplacian. Preprint. arXiv:2409.18744

arXiv
[5]

Quelques aspects non lin´ eaires du principe du maximum

Philippe B´ enilan and Colette Picard. Quelques aspects non lin´ eaires du principe du maximum. In S´ eminaire de Th´ eorie du Potentiel, No. 4 (Paris, 1977/1978), volume 713 of Lecture Notes in Math. , pages 1–37. Springer, Berlin, 1979

work page 1977
[6]

Beurling and J

A. Beurling and J. Deny. Espaces de Dirichlet. I. Le cas ´ e l´ ementaire.Acta Math. , 99:203–224, 1958

work page 1958
[7]

Beurling and J

A. Beurling and J. Deny. Dirichlet spaces. Proc. Nat. Acad. Sci. U.S.A. , 45:208–215, 1959

work page 1959
[8]

No nlinear Dirichlet forms associated with quasiregular mappings

Camelia Beznea, Lucian Beznea, and Michael R¨ ockner. No nlinear Dirichlet forms associated with quasiregular mappings. Potential Anal. , to appear

work page

Show all 40 references

[9]

The normal contr action property for non- bilinear Dirichlet forms

Giovanni Brigati and Ivailo Hartarsky. The normal contr action property for non- bilinear Dirichlet forms. Potential Anal. , 60(1):473–488, 2024

2024
[10]

Reﬂexive Orlicz spaces hav e uniformly normal structure

Shu Tao Chen and Hui Ying Sun. Reﬂexive Orlicz spaces hav e uniformly normal structure. Studia Math. , 109(2):197–208, 1994

1994
[11]

Symmetric Markov processes, time change, and boundary theory , volume 35 of London Mathematical Society Monographs Series

Zhen-Qing Chen and Masatoshi Fukushima. Symmetric Markov processes, time change, and boundary theory , volume 35 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2012

2012
[12]

Geometry of Banach spaces, duality mappings and nonlinear p rob- lems, volume 62 of Mathematics and its Applications

Ioana Cioranescu. Geometry of Banach spaces, duality mappings and nonlinear p rob- lems, volume 62 of Mathematics and its Applications . Kluwer Academic Publishers Group, Dordrecht, 1990. 52 MARCEL SCHMIDT AND IAN ZIMMERMANN

1990
[13]

Nonlinear Markov s emigroups, nonlinear Dirichlet forms and applications to minimal surfaces

Fabio Cipriani and Gabriele Grillo. Nonlinear Markov s emigroups, nonlinear Dirichlet forms and applications to minimal surfaces. J. Reine Angew. Math. , 562:201–235, 2003

2003
[14]

Energy spaces, Dirichlet forms and cap acities in a nonlinear setting

Burkhard Claus. Energy spaces, Dirichlet forms and cap acities in a nonlinear setting. Potential Anal. , 58(1):159–179, 2023

2023
[15]

Nonlinear dirichlet forms

Burkhard Claus. Nonlinear dirichlet forms. Dissertat ion, 2021

2021
[16]

A semigrou p approach to nonlinear L´ evy processes.Stochastic Process

Robert Denk, Michael Kupper, and Max Nendel. A semigrou p approach to nonlinear L´ evy processes.Stochastic Process. Appl. , 130(3):1616–1642, 2020

2020
[17]

Optimal Lp Hardy-type inequalities

Baptiste Devyver and Yehuda Pinchover. Optimal Lp Hardy-type inequalities. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 33(1):93–118, 2016

2016
[18]

Springer, Heidelberg, 2011

Lars Diening, Petteri Harjulehto, Peter H¨ ast¨ o, and M ichael R˚ uˇ ziˇ cka.Lebesgue and Sobolev spaces with variable exponents , volume 2017 of Lecture Notes in Mathematics . Springer, Heidelberg, 2011

2017
[19]

Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, english edition, 1999

Ivar Ekeland and Roger T´ emam.Convex analysis and variational problems , volume 28 of Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, english edition, 1999. Translat ed from the French

1999
[20]

A non-local quasi-linear ground stat e representation and criticality theory

Florian Fischer. A non-local quasi-linear ground stat e representation and criticality theory. Calc. Var. Partial Diﬀerential Equations , 62(5):Paper No. 163, 33, 2023

2023
[21]

Dirichlet forms and symmetric Markov processes, volume 19 of de Gruyter Studies in Mathematics

Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Tak eda. Dirichlet forms and symmetric Markov processes, volume 19 of de Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, extended edition, 2011

2011
[22]

Analytic and geometric background of recurrence and no n- explosion of the Brownian motion on Riemannian manifolds

Alexander Grigor ′yan. Analytic and geometric background of recurrence and no n- explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.), 36(2):135–249, 1999

1999
[23]

Towards an Lp-potential theory for sub-Markovian semigroups: variational inequalities and balayage theory

Walter Hoh and Niels Jacob. Towards an Lp-potential theory for sub-Markovian semigroups: variational inequalities and balayage theory . J. Evol. Equ. , 4(2):297–312, 2004

2004
[24]

The open mapping and closed graph theorems in topological ve ctor spaces

Taqdir Husain. The open mapping and closed graph theorems in topological ve ctor spaces. Clarendon Press, Oxford, 1965

1965
[25]

Schilling

Niels Jacob and Ren´ e L. Schilling. Extended Lp Dirichlet spaces. In Around the research of Vladimir Maz’ya. I , volume 11 of Int. Math. Ser. (N. Y.) , pages 221–238. Springer, New York, 2010

2010
[26]

(1 , p)-Sobolev spaces based on strongly local Dirichlet forms

Kazuhiro Kuwae. (1 , p)-Sobolev spaces based on strongly local Dirichlet forms. Math. Nachr., 297(10):3723–3740, 2024

2024
[27]

Mar kov processes associated with semi-Dirichlet forms

Zhi Ming Ma, Ludger Overbeck, and Michael R¨ ockner. Mar kov processes associated with semi-Dirichlet forms. Osaka J. Math. , 32(1):97–119, 1995

1995
[28]

Introduction to the theory of (nonsymmetric) Dirichlet forms

Zhi Ming Ma and Michael R¨ ockner. Introduction to the theory of (nonsymmetric) Dirichlet forms . Universitext. Springer-Verlag, Berlin, 1992

1992
[29]

Semi-Dirichlet forms and Markov processes , volume 48 of De Gruyter Studies in Mathematics

Yoichi Oshima. Semi-Dirichlet forms and Markov processes , volume 48 of De Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, 2013

2013
[30]

Topics in the theory of positive solu tions of second-order ellip- tic and parabolic partial diﬀerential equations

Yehuda Pinchover. Topics in the theory of positive solu tions of second-order ellip- tic and parabolic partial diﬀerential equations. In Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon ’s 60th birthd ay, volume 76 of Proc. Sympos. Pure Math. , p...

2007
[31]

Ground state alte rnative for p-Laplacian with potential term

Yehuda Pinchover and Kyril Tintarev. Ground state alte rnative for p-Laplacian with potential term. Calc. Var. Partial Diﬀerential Equations , 28(2):179–201, 2007

2007
[32]

Characterization of nonlinear Dirichl et forms via contractions

Simon Puchert. Characterization of nonlinear Dirichl et forms via contractions. Preprint. arXiv:2502.03691

arXiv
[33]

Weak Poincar´ e inequalities and L2-convergence rates of Markov semigroups

Michael R¨ ockner and Feng-Yu Wang. Weak Poincar´ e inequalities and L2-convergence rates of Markov semigroups. J. Funct. Anal. , 185(2):564–603, 2001

2001
[34]

Energy forms

Marcel Schmidt. Energy forms. Dissertation, arXiv:1703.04883 , 2017

2017 arXiv
[35]

A note on reﬂected Dirichlet forms

Marcel Schmidt. A note on reﬂected Dirichlet forms. Potential Anal. , 52(2):245–279, 2020

2020
[36]

(Weak) Hardy and Poincar´ e inequaliti es and criticality theory

Marcel Schmidt. (Weak) Hardy and Poincar´ e inequaliti es and criticality theory. In Dirichlet forms and related topics , volume 394 of Springer Proc. Math. Stat. , pages 421–459. Springer, Singapore, [2022] ©2022. NONLINEAR DIRICHLET FORMS 53

2022
[37]

Positivity preserving forms have the Fatou property

Byron Schmuland. Positivity preserving forms have the Fatou property. Potential Anal., 10(4):373–378, 1999

1999
[38]

R. E. Showalter. Monotone operators in Banach space and nonlinear partial di ﬀeren- tial equations, volume 49 of Mathematical Surveys and Monographs . American Math- ematical Society, Providence, RI, 1997

1997
[39]

Silverstein

Martin L. Silverstein. Symmetric Markov processes . Lecture Notes in Mathematics, Vol. 426. Springer-Verlag, Berlin-New York, 1974

1974
[40]

On nonlinear Dirichlet forms

Petra van Beusekom. On nonlinear Dirichlet forms. PhD Thesis , 1994. Mathematisches Institut, Universit ¨at Leipzig, 04109 Leipzig, Germany. Email address : marcel.schmidt@math.uni-leipzig.de Institut f ¨ur Mathematik, Friedrich-Schiller-Universit ¨at Jena, 07743 Jena, Germany...

1994

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Necessary and suﬃcient conditions for the exis- tence of m-perfect processes associated with Dirichlet forms

Sergio Albeverio and Zhi Ming Ma. Necessary and suﬃcient conditions for the exis- tence of m-perfect processes associated with Dirichlet forms. In S´ eminaire de Proba- bilit´ es, XXV, volume 1485 of Lecture Notes in Math. , pages 374–406. Springer, Berlin, 1991

work page 1991

[2] [2]

A general corresponden ce between Dirichlet forms and right processes

Sergio Albeverio and Zhi Ming Ma. A general corresponden ce between Dirichlet forms and right processes. Bull. Amer. Math. Soc. (N.S.) , 26(2):245–252, 1992

work page 1992

[3] [3]

Cal culus and heat ﬂow in metric measure spaces and applications to spaces with Ricci bounds from below

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´ e. Cal culus and heat ﬂow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math., 195(2):289–391, 2014

work page 2014

[4] [4]

p-brownian motion and the p-laplacian

Viorel Barbu, Marco Rehmeier, and Michael R¨ ockner. p-brownian motion and the p-laplacian. Preprint. arXiv:2409.18744

arXiv

[5] [5]

Quelques aspects non lin´ eaires du principe du maximum

Philippe B´ enilan and Colette Picard. Quelques aspects non lin´ eaires du principe du maximum. In S´ eminaire de Th´ eorie du Potentiel, No. 4 (Paris, 1977/1978), volume 713 of Lecture Notes in Math. , pages 1–37. Springer, Berlin, 1979

work page 1977

[6] [6]

Beurling and J

A. Beurling and J. Deny. Espaces de Dirichlet. I. Le cas ´ e l´ ementaire.Acta Math. , 99:203–224, 1958

work page 1958

[7] [7]

Beurling and J

A. Beurling and J. Deny. Dirichlet spaces. Proc. Nat. Acad. Sci. U.S.A. , 45:208–215, 1959

work page 1959

[8] [8]

No nlinear Dirichlet forms associated with quasiregular mappings

Camelia Beznea, Lucian Beznea, and Michael R¨ ockner. No nlinear Dirichlet forms associated with quasiregular mappings. Potential Anal. , to appear

work page