Harnack inequality for $p$-harmonic functions: improved dimension dependence via tug of war

Han Wang; Yuval Peres

arxiv: 2604.10142 · v2 · submitted 2026-04-11 · 🧮 math.PR · math.AP

Harnack inequality for p-harmonic functions: improved dimension dependence via tug of war

Yuval Peres , Han Wang This is my paper

Pith reviewed 2026-05-12 01:50 UTC · model grok-4.3

classification 🧮 math.PR math.AP

keywords Harnack inequalityp-harmonic functionstug-of-war gamesMoser iterationdimension dependenceprobabilistic methodsnonlinear elliptic equations

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The pith

Tug-of-war games produce a Harnack inequality constant for p-harmonic functions of order exp(C_p d log d) in high dimensions.

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

The paper refines the probabilistic tug-of-war game method to prove the Harnack inequality for p-harmonic functions in bounded domains of R^d. This yields a constant that grows only as exp(C_p d log d) when dimension increases, improving the dependence obtained from the classical Moser iteration technique. A reader would care because these functions govern nonlinear elliptic equations whose regularity controls many problems in analysis and geometry. The improved bound holds uniformly for every p greater than 1.

Core claim

By refining the tug-of-war analysis from Luiro, Parviainen and Saksman, the authors show that the constant in Harnack's inequality for p-harmonic functions is O(exp(C_p d log d)) as d tends to infinity. This improves the constant derived from Moser iteration and applies for all p>1 in bounded domains.

What carries the argument

Refined tug-of-war game with p-dependent noise, which represents p-harmonic functions via expected game payoffs and controls their oscillations through probabilistic estimates on exit times and strategies.

If this is right

Holder continuity estimates for p-harmonic functions inherit the same improved dimension dependence.
The same refined game analysis applies to other inequalities previously derived via tug-of-war for p-harmonic functions.
The bounds require no additional boundary regularity assumptions beyond the domain being bounded.
The result extends the range of dimensions where explicit control on p-harmonic oscillation is available.

Where Pith is reading between the lines

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

The game representation might permit computing or approximating the Harnack constant numerically by simulating optimal strategies in concrete domains.
Analogous refinements could improve constants for other nonlinear operators whose solutions admit game-theoretic interpretations.
In high-dimensional numerical PDE work the reduced growth might make iteration-free methods competitive with Moser-type schemes.

Load-bearing premise

The refined tug-of-war estimates from 2013 carry over to every bounded domain in R^d and every p>1 without extra restrictions on the domain or its boundary.

What would settle it

An explicit example of a p-harmonic function in a high-dimensional domain where the supremum-to-infimum ratio at two interior points exceeds any multiple of exp(C d log d) times the boundary data oscillation.

Figures

Figures reproduced from arXiv: 2604.10142 by Han Wang, Yuval Peres.

**Figure 1.** Figure 1: In the left figure, when Un+1 is in the direction very close to Ẑn, the strategy S ∗ II tries to cancel this intended move. In the right figure, when Un+1 is not in the direction Ẑn, the strategy just moves towards Yn. Given the current position Xn, Yn and the move Un+1 where player I would move according to SI , the counter-strategy S ∗ II chooses a vector ⎧⎪⎪ ⎨ ⎪⎪⎩ −Un+1, if Ẑn ● Ûn+1 > cos θ0; −εẐn… view at source ↗

read the original abstract

Let $p>1$. The Harnack inequality and H\"older continuity for $p$-harmonic functions in bounded domains in $\mathbb{R}^d$ are usually proved via Moser iteration. In 2013 Luiro, Parviainen and Saksman showed that tug-of-war games can also be used to derive these inequalities. We refine their analysis and obtain improved dependence on $p$ and the dimension $d$ by probabilistic methods. In particular, we show that for all $p>1$, the constant in Harnack's inequality is $O(\exp(C_p d\log d))$ as $d\rightarrow\infty$, which improves the constant derived from Moser iteration.

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

Peres and Wang refine the 2013 tug-of-war analysis to get an O(exp(C_p d log d)) Harnack constant for p-harmonics, beating the Moser iteration dependence on dimension.

read the letter

The main takeaway is that this paper sharpens the dimension scaling in the Harnack inequality for p-harmonic functions. By refining the probabilistic estimates in the tug-of-war game from Luiro-Parviainen-Saksman 2013, they reach a constant of order exp(C_p d log d) as d grows, which is better than what comes out of Moser iteration for the same setting in bounded domains and all p > 1. That is the concrete advance the abstract highlights, and it looks like a direct, incremental improvement rather than a wholesale change in approach. The probabilistic method lets them track the growth more cleanly than the iterative Moser scheme, which is the part that works well here. It gives an alternative route that might be easier to adapt in some high-d contexts where the Moser constants get cumbersome. The claim stays within standard assumptions on the domain and the p-Laplacian, with no obvious extra restrictions introduced. On the soft spots, the bound remains exponential in d, just with a milder log d factor inside, so the practical gain for very large d is limited even if the math is cleaner. The abstract does not include error estimates or a side-by-side constant comparison, which means the full paper needs to be checked to confirm how much the refinement actually buys and whether C_p stays reasonable as p varies. Nothing in the setup suggests circularity or hidden fitting. This is targeted at researchers already working on stochastic games for nonlinear PDE or on dimension dependence in elliptic regularity. A reader who follows the tug-of-war literature or wants probabilistic proofs for p-harmonic estimates will find the explicit bound useful. The work is grounded enough in an established method that it deserves a serious referee to verify the details of the tightened estimates rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The manuscript refines the tug-of-war game analysis from Luiro-Parviainen-Saksman (2013) to establish the Harnack inequality and Hölder continuity for p-harmonic functions on bounded domains in R^d. The central result is an improved dimension dependence: for every fixed p>1 the Harnack constant is O(exp(C_p d log d)) as d→∞, which is better than the dependence obtained from Moser iteration.

Significance. If the claimed bound holds, the improvement is significant because it supplies a sharper, explicitly dimension-dependent constant for a fundamental inequality in nonlinear potential theory via probabilistic methods. The work builds directly on the 2013 tug-of-war framework without introducing new free parameters beyond the p-dependent factor C_p, and the resulting estimate is falsifiable by direct comparison with Moser constants.

minor comments (2)

[Abstract] The abstract states that the refinement yields improved dependence on both p and d, yet the explicit big-O bound is stated only for d (with C_p). A short paragraph in the introduction or §1 comparing the new constant quantitatively to the Moser-iteration constant would make the improvement fully transparent.
The weakest assumption listed in the analysis—that the refined tug-of-war estimates hold in arbitrary bounded domains without extra boundary regularity—should be stated explicitly as a hypothesis in the main theorem statement so that readers can immediately see the precise setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The paper refines the independent 2013 tug-of-war analysis of Luiro-Parviainen-Saksman (different authors) via probabilistic methods to obtain the Harnack constant bound O(exp(C_p d log d)). No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain. The central claim is derived from game estimates in bounded domains and does not equate to its inputs by construction. This is the expected non-finding for a refinement paper whose probabilistic estimates are presented as independent of the target constant.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the probabilistic representation of p-harmonic functions via tug-of-war games established in prior work, plus the technical refinement of game strategies for dimension control.

free parameters (1)

C_p
C_p is an unspecified constant depending on p that appears in the exponential bound; its explicit form or derivation is not provided in the abstract.

axioms (1)

domain assumption p-harmonic functions coincide with the value functions of suitably defined tug-of-war games in bounded domains
This is the foundational connection taken from the 2013 Luiro-Parviainen-Saksman paper that the current work refines.

pith-pipeline@v0.9.0 · 5409 in / 1471 out tokens · 115928 ms · 2026-05-12T01:50:31.085994+00:00 · methodology

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Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

DiBenedetto

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A. Harnack. Die Grundlagen der Theorie des logarithmischen Potentiale s und der eindeutigen Potentialfunktion in der Ebene . B. G. Teubner, Leipzig, 1887

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[3]

Heinonen, T

J. Heinonen, T. Kilpel¨ ainen, and O. Martio. Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. The Clarendon Press, Oxf ord University Press, New York, 1993. Oxford Science Publications

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Juutinen, P

P. Juutinen, P. Lindqvist, and J. J. Manfredi. On the equi valence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. , 33(3):699–717, 2001

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[5]

J. L. Lewis. Regularity of the derivatives of solutions t o certain degenerate elliptic equations. Indiana Univ. Math. J. , 32(6):849–858, 1983

work page 1983
[6]

Lindqvist

P. Lindqvist. Notes on the stationary p-Laplace equation . SpringerBriefs in Mathematics. Springer, Cham, 2019

work page 2019
[7]

Luiro and M

H. Luiro and M. Parviainen. Regularity for nonlinear sto chastic games. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 35(6):1435–1456, 2018

work page 2018
[8]

Luiro, M

H. Luiro, M. Parviainen, and E. Saksman. Harnack’s inequ ality for p-harmonic functions via stochastic games. Comm. Partial Diﬀerential Equations , 38(11):1985–2003, 2013

work page 1985
[9]

J. Moser. On Harnack’s theorem for elliptic diﬀerential equations. Comm. Pure Appl. Math. , 14:577–591, 1961

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[10]

J. Moser. A Harnack inequality for parabolic diﬀerenti al equations. Comm. Pure Appl. Math. , 17:101–134, 1964

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[11]

Parviainen

M. Parviainen. Notes on tug-of-war games and the p-Laplace equation. SpringerBriefs on PDEs and Data Science. Springer, Singapore, 2024

work page 2024
[12]

Peres and S

Y. Peres and S. Sheﬃeld. Tug-of-war with noise: a game-t heoretic view of the p-Laplacian. Duke Math. J. , 145(1):91–120, 2008

work page 2008
[13]

J. Serrin. On the Harnack inequality for linear ellipti c equations. J. Analyse Math. , 4:292–308, 1955

work page 1955
[14]

N. S. Trudinger. On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. , 20:721–747, 1967

work page 1967
[15]

N. N. Uraltseva. Degenerate quasilinear elliptic syst ems. Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) , 7:184–222, 1968. Y. Peres, Beijing Institute of Mathematical Sciences and Ap plications, Beijing, China Email address : yperes@bimsa.cn H. W ang, Qiuzhen College, Tsinghua University, Beijing, Ch ina Email address : wanghan21@mails....

work page 1968

[1] [1]

DiBenedetto

E. DiBenedetto. C 1+ α local regularity of weak solutions of degenerate elliptic e quations. Non- linear Anal. , 7(8):827–850, 1983

work page 1983

[2] [2]

A. Harnack. Die Grundlagen der Theorie des logarithmischen Potentiale s und der eindeutigen Potentialfunktion in der Ebene . B. G. Teubner, Leipzig, 1887

work page

[3] [3]

Heinonen, T

J. Heinonen, T. Kilpel¨ ainen, and O. Martio. Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. The Clarendon Press, Oxf ord University Press, New York, 1993. Oxford Science Publications

work page 1993

[4] [4]

Juutinen, P

P. Juutinen, P. Lindqvist, and J. J. Manfredi. On the equi valence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. , 33(3):699–717, 2001

work page 2001

[5] [5]

J. L. Lewis. Regularity of the derivatives of solutions t o certain degenerate elliptic equations. Indiana Univ. Math. J. , 32(6):849–858, 1983

work page 1983

[6] [6]

Lindqvist

P. Lindqvist. Notes on the stationary p-Laplace equation . SpringerBriefs in Mathematics. Springer, Cham, 2019

work page 2019

[7] [7]

Luiro and M

H. Luiro and M. Parviainen. Regularity for nonlinear sto chastic games. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 35(6):1435–1456, 2018

work page 2018

[8] [8]

Luiro, M

H. Luiro, M. Parviainen, and E. Saksman. Harnack’s inequ ality for p-harmonic functions via stochastic games. Comm. Partial Diﬀerential Equations , 38(11):1985–2003, 2013

work page 1985

[9] [9]

J. Moser. On Harnack’s theorem for elliptic diﬀerential equations. Comm. Pure Appl. Math. , 14:577–591, 1961

work page 1961

[10] [10]

J. Moser. A Harnack inequality for parabolic diﬀerenti al equations. Comm. Pure Appl. Math. , 17:101–134, 1964

work page 1964

[11] [11]

Parviainen

M. Parviainen. Notes on tug-of-war games and the p-Laplace equation. SpringerBriefs on PDEs and Data Science. Springer, Singapore, 2024

work page 2024

[12] [12]

Peres and S

Y. Peres and S. Sheﬃeld. Tug-of-war with noise: a game-t heoretic view of the p-Laplacian. Duke Math. J. , 145(1):91–120, 2008

work page 2008

[13] [13]

J. Serrin. On the Harnack inequality for linear ellipti c equations. J. Analyse Math. , 4:292–308, 1955

work page 1955

[14] [14]

N. S. Trudinger. On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. , 20:721–747, 1967

work page 1967

[15] [15]

N. N. Uraltseva. Degenerate quasilinear elliptic syst ems. Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) , 7:184–222, 1968. Y. Peres, Beijing Institute of Mathematical Sciences and Ap plications, Beijing, China Email address : yperes@bimsa.cn H. W ang, Qiuzhen College, Tsinghua University, Beijing, Ch ina Email address : wanghan21@mails....

work page 1968