A Projected Tug-of-War Game for the Regularized $p$-Laplacian

Behrooz Moosavi Ramezanzadeh

arxiv: 2605.01218 · v1 · submitted 2026-05-02 · 🧮 math.AP

A Projected Tug-of-War Game for the Regularized p-Laplacian

Behrooz Moosavi Ramezanzadeh This is my paper

Pith reviewed 2026-05-09 18:32 UTC · model grok-4.3

classification 🧮 math.AP

keywords regularized p-Laplaciantug-of-war gamesp-harmonious functionsviscosity solutionsdynamic programming principlelinear liftprojection

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The pith

Projecting the p-harmonious tug-of-war scheme from one higher dimension produces a discrete dynamic programming principle whose solutions converge to the viscosity solution of the regularized p-Laplacian.

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

The paper establishes a game-theoretic representation for the regularized p-Laplacian equation div((1 + |Dv|^2)^{p/2-1} Dv) = 0. It uses a linear lift that turns this equation into the standard p-Laplacian in one extra dimension. The known tug-of-war scheme in n+1 dimensions is projected down to n dimensions, creating a new discrete dynamic programming principle. For boundary data that does not depend on the extra coordinate, the authors prove existence and uniqueness of a Borel measurable solution to this principle, show that it equals the value of the projected game, and prove that the solutions approach the viscosity solution of the original equation as the discretization parameter tends to zero. A reader would care because the construction supplies an explicit discrete approximation scheme grounded in a probabilistic game for a nonlinear elliptic equation.

Core claim

Via the linear lift w(x, x_{n+1}) = v(x) + x_{n+1}, the regularized p-Laplacian in a bounded domain of R^n is identified with the p-Laplacian in R^{n+1}. Projecting the standard (n+1)-dimensional p-harmonious scheme onto R^n yields a discrete dynamic programming principle. For strip boundary data, this principle possesses a unique Borel measurable solution that coincides with the value of the projected game and converges to the viscosity solution of the regularized equation as ε → 0.

What carries the argument

the linear lift w(x, x_{n+1}) = v(x) + x_{n+1} that converts the regularized p-Laplacian into the standard p-Laplacian, together with the projection of the (n+1)-dimensional p-harmonious scheme onto a discrete dynamic programming principle in n dimensions

Load-bearing premise

The linear lift must correctly transform the regularized p-Laplacian into the standard p-Laplacian in one higher dimension, and the boundary data must be of strip type independent of the extra coordinate.

What would settle it

A numerical test on the unit disk with constant boundary data that checks whether the unique fixed point of the projected discrete scheme fails to approach the known explicit viscosity solution of the regularized p-Laplacian for some p > 2 as the step size ε is driven to zero.

read the original abstract

We give a tug-of-war interpretation of the regularized $p$-Laplacian $\divgg\big((1+|Dv|^2)^{p/2-1}Dv\big)=0$ in a bounded domain $\Omega\subset\R^n$, $p\ge 2$. The key is the linear lift $w(x,x_{n+1})=v(x)+x_{n+1}$, which identifies this equation with $\Delta_p w=0$ in $\R^{n+1}$. Projecting the standard $(n+1)$-dimensional $p$-harmonious scheme onto $\R^n$ yields a discrete dynamic programming principle for which we prove existence, uniqueness, and Borel measurability of solutions with strip boundary data, identify the unique fixed point with the value of the projected game, and establish convergence to the viscosity solution as $\varepsilon\to 0$.

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 gives a clean projection of the tug-of-war scheme onto the regularized p-Laplacian via linear lift, with proofs of well-posedness and convergence for the discrete problem.

read the letter

The main thing here is the projection step. They lift v to w(x, x_{n+1}) = v(x) + x_{n+1} so the regularized p-Laplacian becomes the ordinary p-Laplacian in one higher dimension, then pull the standard (n+1)-dimensional p-harmonious scheme back down to a discrete dynamic programming principle on the original bounded domain with strip boundary data. They prove existence, uniqueness, Borel measurability of the fixed point, identify it with the game value, and show convergence to the viscosity solution as epsilon goes to zero. That package is new in the literature they cite. The lift itself is a simple calculation that works because the extra divergence term drops out, and the projection keeps everything inside Omega. The arguments rest on standard techniques for p-harmonious functions, which is appropriate and keeps the paper focused. The proofs look reproducible in principle once the details are checked. The stress-test concern about finite exit time and linear growth in the extra coordinate is reasonable to raise, since the usual tug-of-war theory uses bounded domains. The paper claims to establish the properties anyway for strip data, so the arguments must adapt the exit-time estimates or value bounds in some way. That step is the one that would need the closest look in the full text, but it does not appear to be a contradiction with the stated results. This is aimed at specialists in viscosity solutions and game-theoretic approximations to degenerate elliptic equations. Someone already working on p-harmonious functions or discrete schemes for the p-Laplacian would get a usable new technique from it. It deserves a serious referee because the claims are specific and the method is a direct, checkable extension of known tools.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a projected tug-of-war game interpretation for the regularized p-Laplacian div((1+|Dv|^2)^{p/2-1} Dv)=0 in a bounded domain Ω⊂R^n (p≥2). Using the linear lift w(x,x_{n+1})=v(x)+x_{n+1}, the equation is identified with the standard p-Laplacian in the cylinder Ω×R. The standard (n+1)-dimensional p-harmonious scheme is projected onto R^n to obtain a discrete dynamic programming principle; the authors prove existence, uniqueness, and Borel measurability of solutions for strip boundary data, identify the unique fixed point with the value of the projected game, and show convergence to the viscosity solution as ε→0.

Significance. If the arguments hold, the work supplies a discrete, game-theoretic approximation scheme for the regularized p-Laplacian that inherits properties from the well-studied p-harmonious functions while handling the regularization via projection. This could facilitate numerical methods and probabilistic representations for a class of quasilinear equations, extending existing tug-of-war literature.

major comments (2)

[proof of existence/uniqueness for the projected DPP and game-value identification] The central well-posedness claims for the projected DPP (existence, uniqueness, Borel measurability, and identification with the game value) rest on inheriting finite exit-time and finite-value properties from the standard theory. However, the strip boundary data g(x)+x_{n+1} on the unbounded cylinder introduces linear growth in the extra coordinate; the manuscript must explicitly verify that the exit time remains almost surely finite and that the value function remains finite (or provide a truncation argument) rather than assuming the bounded-domain theory carries over directly.
[convergence section] In the convergence argument as ε→0, the passage from the discrete projected scheme to the viscosity solution of the regularized equation relies on the lift being exactly harmonic for the standard p-Laplacian. Any error introduced by the projection step (e.g., in the definition of the projected transition probabilities) must be controlled uniformly; the manuscript should quantify this error explicitly rather than invoking the standard convergence theorem verbatim.

minor comments (1)

[preliminaries] The notation for the projected dynamic programming principle should be introduced with a clear display equation immediately after the lift is defined, to avoid ambiguity between the (n+1)-dimensional and projected operators.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight important points regarding the well-posedness and convergence arguments, which we address point by point below. We plan to incorporate clarifications and additional details in the revised version.

read point-by-point responses

Referee: [proof of existence/uniqueness for the projected DPP and game-value identification] The central well-posedness claims for the projected DPP (existence, uniqueness, Borel measurability, and identification with the game value) rest on inheriting finite exit-time and finite-value properties from the standard theory. However, the strip boundary data g(x)+x_{n+1} on the unbounded cylinder introduces linear growth in the extra coordinate; the manuscript must explicitly verify that the exit time remains almost surely finite and that the value function remains finite (or provide a truncation argument) rather than assuming the bounded-domain theory carries over directly.

Authors: We agree that the linear growth in the extra coordinate requires explicit verification rather than direct inheritance from bounded-domain results. In the manuscript, the proofs of existence, uniqueness, and game-value identification for the projected DPP adapt the standard p-harmonious arguments by projecting the (n+1)-dimensional scheme, but we acknowledge the need for a dedicated check. In the revision, we will add a lemma or subsection explicitly proving that the exit time from the cylinder is almost surely finite (using the positive probability of exiting the bounded base domain Ω in finite steps under the projected random walk) and that the value function is finite (by bounding the expected payoff via the linear growth term and the finite moments of the exit time). This will be inserted prior to the main well-posedness theorem without changing the overall strategy. revision: yes
Referee: [convergence section] In the convergence argument as ε→0, the passage from the discrete projected scheme to the viscosity solution of the regularized equation relies on the lift being exactly harmonic for the standard p-Laplacian. Any error introduced by the projection step (e.g., in the definition of the projected transition probabilities) must be controlled uniformly; the manuscript should quantify this error explicitly rather than invoking the standard convergence theorem verbatim.

Authors: The referee correctly identifies that the projection of the transition probabilities could in principle introduce a discrepancy. However, the construction ensures that for the linear lift the projected operator reproduces the exact p-harmonious expectation in the extra coordinate, so the error is not arbitrary. To make the argument fully rigorous, we will revise the convergence section to include an explicit uniform estimate on the difference between the projected DPP operator and the standard (n+1)-dimensional operator; this difference is shown to be O(ε) (or smaller) uniformly on compact sets, which vanishes in the viscosity limit. With this control in place, the standard convergence theorem for p-harmonious functions can then be applied to the lifted functions, yielding the desired convergence for the original regularized equation. revision: yes

Circularity Check

0 steps flagged

No circularity: proofs of well-posedness for projected DPP rely on independent arguments

full rationale

The derivation begins with the linear lift w(x, x_{n+1}) = v(x) + x_{n+1} that directly equates the regularized p-Laplacian to the standard p-Laplacian in one higher dimension via explicit computation of |Dw|^2 and the divergence term. Projection of the (n+1)-dimensional p-harmonious scheme then produces a discrete DPP on the bounded domain Ω with strip data; the paper states that it proves existence, uniqueness, Borel measurability of the fixed point, identification with the game value, and ε→0 convergence to the viscosity solution. These steps are presented as theorems established by standard dynamic-programming and viscosity techniques rather than by re-using the target equation or fitted parameters. No quoted reduction equates any claimed result to its own inputs by construction, and no load-bearing premise collapses to a self-citation chain. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical background in viscosity solutions and game theory with no free parameters or new postulated entities.

axioms (2)

standard math Standard comparison principles and existence theory for viscosity solutions of p-Laplacian-type equations
The convergence argument invokes viscosity solution theory which depends on these background results.
domain assumption Existence and uniqueness of p-harmonious functions for the standard p-Laplacian in higher dimensions
The projection step builds directly on known results for the unregularized p-Laplacian in R^{n+1}.

pith-pipeline@v0.9.0 · 5447 in / 1463 out tokens · 40337 ms · 2026-05-09T18:32:17.892528+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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Manfredi, and Xiaodan Zhou,Superposition property in disjoint variables for the infinity Laplace equation, 2025, Preprint

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Manfredi, Mikko Parviainen, and Julio D

Juan J. Manfredi, Mikko Parviainen, and Julio D. Rossi,On the definition and properties ofp- harmonious functions, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze11(2012), no. 2, 215–241

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Manfredi, and Mikko Parviainen,Connections between coupling and Ishii–Lions methods for tug-of-war with noise stochastic games, 2025, Preprint

Riku Anttila, Juan J. Manfredi, and Mikko Parviainen,Connections between coupling and Ishii–Lions methods for tug-of-war with noise stochastic games, 2025, Preprint

work page 2025

[2] [2]

Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions,User’s guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society27(1992), no. 1, 1–67

work page 1992

[3] [3]

48, Springer-Verlag, Berlin, 1969

Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea,Topics in the theory of lifting, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48, Springer-Verlag, Berlin, 1969

work page 1969

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Manfredi, and Xiaodan Zhou,Superposition property in disjoint variables for the infinity Laplace equation, 2025, Preprint

Qing Liu, Juan J. Manfredi, and Xiaodan Zhou,Superposition property in disjoint variables for the infinity Laplace equation, 2025, Preprint

work page 2025

[5] [5]

3–4, 201–216

HannesLuiro, MikkoParviainen, andEeroSaksman,On the existence and uniqueness ofp-harmonious functions, Differential and Integral Equations27(2014), no. 3–4, 201–216

work page 2014

[6] [6]

Manfredi, Mikko Parviainen, and Julio D

Juan J. Manfredi, Mikko Parviainen, and Julio D. Rossi,On the definition and properties ofp- harmonious functions, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze11(2012), no. 2, 215–241

work page 2012

[7] [7]

1, 91–120

Yuval Peres and Scott Sheffield,Tug-of-war with noise: a game-theoretic view of thep-laplacian, Duke Mathematical Journal145(2008), no. 1, 91–120

work page 2008