Energy integrals and asymmetric co-potentials for closed forms

Kaneharu Tsuchida; Kazuhiro Kuwae; Takumu Ooi; Toshihiro Uemura

arxiv: 2607.00853 · v1 · pith:L465HXDWnew · submitted 2026-07-01 · 🧮 math.PR · math.FA

Energy integrals and asymmetric co-potentials for closed forms

Kazuhiro Kuwae , Takumu Ooi , Kaneharu Tsuchida , Toshihiro Uemura This is my paper

Pith reviewed 2026-07-02 07:00 UTC · model grok-4.3

classification 🧮 math.PR math.FA

keywords non-symmetric closed formsfinite energy integralspotentialsco-potentialsDirichlet formsStollmann-Voigt inequalityjump processes

0 comments

The pith

Non-symmetric closed forms require separate analysis for energy integrals, potentials and co-potentials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines measures of finite energy integrals and the associated potentials and co-potentials for non-symmetric closed forms. It performs comparisons against the symmetric case from three angles: a non-symmetric version of the Stollmann-Voigt inequality, non-symmetric perturbations of symmetric forms, and forms arising from non-symmetric jump-type processes. The comparisons establish that these objects behave differently when symmetry is dropped. This shows that the non-symmetric setting needs more careful handling than the symmetric one.

Core claim

Measures of finite energy integrals, potentials, and co-potentials associated with non-symmetric closed forms behave differently from their symmetric counterparts, necessitating more delicate analysis as demonstrated through three viewpoints: a non-symmetric Stollmann-Voigt inequality, non-symmetric perturbations, and closed forms from non-symmetric jump-type processes.

What carries the argument

Comparison of finite energy integral measures and asymmetric co-potentials under non-symmetric closed forms via the non-symmetric Stollmann-Voigt inequality, perturbations, and jump-type forms.

If this is right

The non-symmetric Stollmann-Voigt inequality produces distinct bounds on energy integrals compared with the symmetric version.
Perturbations of symmetric forms by non-symmetric terms yield potentials and co-potentials that cannot be read off from the symmetric theory.
Closed forms coming from non-symmetric jump-type processes require independent verification of finite energy integral properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The observed differences may affect how one constructs resolvents or semigroups for processes with non-symmetric generators.
Concrete examples such as diffusions with drift could be used to quantify the size of the gap between symmetric and non-symmetric energy integrals.

Load-bearing premise

The non-symmetric closed forms under consideration admit well-defined measures of finite energy integrals that allow the stated comparisons via the three viewpoints.

What would settle it

An explicit non-symmetric closed form in which the measures of finite energy integrals and the co-potentials coincide exactly with the symmetric case would show that the claimed difference does not hold in general.

read the original abstract

We investigate the class of measures of finite energy integrals and the behavior of potentials and co-potentials associated with non-symmetric closed forms. In particular, we compare these objects with their symmetric counterparts from three viewpoints: a non-symmetric version of Stollmann--Voigt's inequality, non-symmetric perturbations of symmetric forms, and closed forms associated with non-symmetric jump-type forms. Our results indicate that measures of finite energy integrals, potentials, and co-potentials behave differently in the non-symmetric setting, requiring more delicate analysis than in the symmetric case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Stollmann-Voigt and related comparisons for energy integrals and potentials to non-symmetric closed forms via three viewpoints, but the abstract gives no sense of how substantial the differences are.

read the letter

The new material is the non-symmetric Stollmann-Voigt inequality together with the comparisons obtained from perturbations of symmetric forms and from jump-type forms. These give concrete ways to see that measures of finite energy integrals, potentials, and co-potentials do not carry over unchanged when symmetry is removed.

The paper does a clean job of organizing the three viewpoints and stating that the non-symmetric objects need more delicate handling. That matches what one would expect from the literature on non-symmetric Dirichlet forms.

The soft spot is that the abstract only reports the existence of differences without showing the actual statements, any quantitative gap, or even a single explicit example. Without the full text it is impossible to judge whether the extra delicacy changes how one solves problems or is mostly bookkeeping. The claim that the behavior is different is plausible but remains unquantified here.

This is for specialists who already know the symmetric theory and want the non-symmetric extensions filled in. A reader outside that niche will not get much. The work looks like a standard incremental step in the area with no obvious circularity or invented objects.

I would send it to peer review so the proofs and the size of the differences can be checked by people who work in this subfield.

Referee Report

0 major / 1 minor

Summary. The manuscript investigates measures of finite energy integrals and the behavior of potentials and co-potentials for non-symmetric closed forms. It compares these to symmetric counterparts via three viewpoints: a non-symmetric Stollmann--Voigt inequality, non-symmetric perturbations of symmetric forms, and closed forms from non-symmetric jump-type forms. The central claim is that these objects behave differently in the non-symmetric setting and require more delicate analysis than in the symmetric case.

Significance. If the derivations hold, the work contributes to non-symmetric Dirichlet form theory by identifying distinctions in energy measures and potentials that do not appear in the symmetric setting. The provision of three independent viewpoints (non-symmetric Stollmann--Voigt inequality, perturbations, and jump-type forms) is a strength, as it allows cross-verification of the claimed differences and supplies concrete tools for future analysis in asymmetric potential theory.

minor comments (1)

[Abstract] Abstract: the phrasing is appropriately cautious, but the introduction would benefit from an explicit statement of the main theorems (e.g., the precise form of the non-symmetric Stollmann--Voigt inequality) to orient the reader before the three viewpoints are developed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a theoretical investigation of measures of finite energy integrals, potentials, and co-potentials for non-symmetric closed forms. It compares these objects to symmetric counterparts using a non-symmetric Stollmann--Voigt inequality, perturbations, and jump-type forms. No load-bearing step reduces by construction to a self-definition, fitted input renamed as prediction, or self-citation chain. The central claims rest on external prior theory in Dirichlet forms and potential theory, with the abstract and structure indicating independent mathematical content rather than tautological renaming or ansatz smuggling. This is the expected outcome for a pure math.PR manuscript without empirical fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5621 in / 984 out tokens · 29095 ms · 2026-07-02T07:00:51.655702+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 3 canonical work pages · 1 internal anchor

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M. Fukushima, Y. Oshima and M. Takeda,Dirichlet Forms and Symmetric Markov Pro- cesses, 2nd revised and extended ed., De Gruyter, 2011

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Fukushima and T

M. Fukushima and T. Uemura, Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms,Ann. Probab.,40(2012), 858–889

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Jonsson and H

A. Jonsson and H. Wallin,Function Spaces on Subsets ofR n, Harwood Academic Publish- ers, 1984

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Kuwae, T

K. Kuwae, T. Ooi, K. Tsuchida and T. Uemura,Convergence of positive continuous addi- tive functionals for lower bounded semi-Dirichlet forms, (2026) in preparation

2026
[9]

Nishimori, M

Y. Nishimori, M. Tomisaki, K. Tsuchida and T. Uemura, On a convergence of positive continuous additive functionals in terms of the smooth measures,Canadian J. Math., Published Online, 1–30, (2025), https://doi.org/10.4153/S0008414X25000173

work page doi:10.4153/s0008414x25000173 2025
[10]

T. Ooi, K. Tsuchida and T. Uemura, Smooth measures and positive continuous additive functionals attached to a compact nest, arXiv:2509.23060, (preprint, 2025)

work page arXiv 2025
[11]

T. Ooi, K. Tsuchida and T. Uemura, Classification and metrization of classes of smooth measures, arXiv:2605.05864, (preprint, 2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[12]

Oshima,Semi-Dirichlet Forms and Markov Processes, De Gruyter, 2013

Y. Oshima,Semi-Dirichlet Forms and Markov Processes, De Gruyter, 2013

2013
[13]

Shiozawa and M

Y. Shiozawa and M. Takeda, Variational formula for Dirichlet forms and estimates of principal eigenvalues for symmetricα-stable processes,Potential Anal.23(2005), 135– 151. 29

2005
[14]

R. L. Schilling and J. Wang, Lower bounded semi-Dirichlet forms associated with L´ evy type operators, inFestschrift Masatoshi Fukushima(Interdiscip. Math. Sci.,17), 2015, World Scientific, 507–526

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[15]

Stollmann and J

P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures,Potential Anal.5 (1996), 109–138

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Uemura, Non-symmetric perturbation of a lower bounded closed form, (preprint, 2025)

T. Uemura, Non-symmetric perturbation of a lower bounded closed form, (preprint, 2025). 30

2025

[1] [1]

Aizenman and B

M. Aizenman and B. Simon, Brownian motion and Harnack’s inequality for Schr¨ odinger operators,Communications on Pure and Applied Mathematics,35(1982), 209–273

1982

[2] [2]

Bass, Regularity results for stable-like operators,Journal of Functional Analysis,257 (2009), 2693–2722

R.F. Bass, Regularity results for stable-like operators,Journal of Functional Analysis,257 (2009), 2693–2722

2009

[3] [3]

Ben Amor, Trace inequalities for operators associated to regular Dirichlet forms,Forum Math.,16(2004), 417–429

A. Ben Amor, Trace inequalities for operators associated to regular Dirichlet forms,Forum Math.,16(2004), 417–429

2004

[4] [4]

Fitzsimmons, Hardy’s inequality for Dirichlet forms,Journal of Mathematical Analysis and Applications,250(2000), 548–560

P.J. Fitzsimmons, Hardy’s inequality for Dirichlet forms,Journal of Mathematical Analysis and Applications,250(2000), 548–560

2000

[5] [5]

Fukushima, Y

M. Fukushima, Y. Oshima and M. Takeda,Dirichlet Forms and Symmetric Markov Pro- cesses, 2nd revised and extended ed., De Gruyter, 2011

2011

[6] [6]

Fukushima and T

M. Fukushima and T. Uemura, Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms,Ann. Probab.,40(2012), 858–889

2012

[7] [7]

Jonsson and H

A. Jonsson and H. Wallin,Function Spaces on Subsets ofR n, Harwood Academic Publish- ers, 1984

1984

[8] [8]

Kuwae, T

K. Kuwae, T. Ooi, K. Tsuchida and T. Uemura,Convergence of positive continuous addi- tive functionals for lower bounded semi-Dirichlet forms, (2026) in preparation

2026

[9] [9]

Nishimori, M

Y. Nishimori, M. Tomisaki, K. Tsuchida and T. Uemura, On a convergence of positive continuous additive functionals in terms of the smooth measures,Canadian J. Math., Published Online, 1–30, (2025), https://doi.org/10.4153/S0008414X25000173

work page doi:10.4153/s0008414x25000173 2025

[10] [10]

T. Ooi, K. Tsuchida and T. Uemura, Smooth measures and positive continuous additive functionals attached to a compact nest, arXiv:2509.23060, (preprint, 2025)

work page arXiv 2025

[11] [11]

T. Ooi, K. Tsuchida and T. Uemura, Classification and metrization of classes of smooth measures, arXiv:2605.05864, (preprint, 2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[12] [12]

Oshima,Semi-Dirichlet Forms and Markov Processes, De Gruyter, 2013

Y. Oshima,Semi-Dirichlet Forms and Markov Processes, De Gruyter, 2013

2013

[13] [13]

Shiozawa and M

Y. Shiozawa and M. Takeda, Variational formula for Dirichlet forms and estimates of principal eigenvalues for symmetricα-stable processes,Potential Anal.23(2005), 135– 151. 29

2005

[14] [14]

R. L. Schilling and J. Wang, Lower bounded semi-Dirichlet forms associated with L´ evy type operators, inFestschrift Masatoshi Fukushima(Interdiscip. Math. Sci.,17), 2015, World Scientific, 507–526

2015

[15] [15]

Stollmann and J

P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures,Potential Anal.5 (1996), 109–138

1996

[16] [16]

Uemura, Non-symmetric perturbation of a lower bounded closed form, (preprint, 2025)

T. Uemura, Non-symmetric perturbation of a lower bounded closed form, (preprint, 2025). 30

2025