A game-theoretical interpretation for a doubly nonlinear parabolic equation

Carlos Fuertes-Moran; Felix del Teso; Julio D. Rossi

arxiv: 2604.11592 · v1 · submitted 2026-04-13 · 🧮 math.AP · math.PR

A game-theoretical interpretation for a doubly nonlinear parabolic equation

Felix del Teso , Carlos Fuertes-Moran , Julio D. Rossi This is my paper

Pith reviewed 2026-05-10 15:15 UTC · model grok-4.3

classification 🧮 math.AP math.PR

keywords doubly nonlinear parabolic equationp-Laplacianasymptotic mean value formuladynamic programming principlestochastic gamesviscosity solutionstwo-player gameszero-sum games

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The pith

A new asymptotic mean value formula for the p-Laplacian defines a dynamic programming principle whose solutions match viscosity solutions and arise as values of a two-player stochastic game.

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

The paper constructs a game-theoretic interpretation for the doubly nonlinear parabolic equation involving the p-Laplacian with p greater than 2. It begins by deriving a new asymptotic mean value formula that remains valid at points where the spatial gradient vanishes and does not depend on the sign of the p-Laplacian. This formula produces a dynamic programming principle whose solutions converge to the viscosity solution of the associated boundary value problem. The same solutions are shown to be the value functions of a stochastic two-player zero-sum game introduced in the work. A reader would care because the result supplies both a probabilistic representation and a potential computational pathway for this class of nonlinear evolution equations.

Core claim

We introduce a game-theoretical framework for the doubly nonlinear parabolic equation |∂_t u|^{p-2} ∂_t u - Δ_p u = 0. A key feature to our approach is a new asymptotic mean value formula for the p-Laplacian that is robust even when the gradient vanishes and is independent of the sign of the p-Laplacian. This new AMVF leads naturally to a dynamic programming principle whose solutions converge to the viscosity solution of the boundary value problem for the differential equation. In addition, solutions to the DPP coincide with value functions for a stochastic, two-players, zero-sum game that we introduce and analyze here.

What carries the argument

The new asymptotic mean value formula for the p-Laplacian, which remains valid at vanishing gradients and independent of sign, and which directly yields both the dynamic programming principle and the stochastic game.

If this is right

Solutions of the dynamic programming principle converge to the viscosity solution of the boundary value problem.
Solutions of the dynamic programming principle are exactly the value functions of the stochastic two-player zero-sum game.
The game supplies a probabilistic representation of solutions to the doubly nonlinear equation.
The construction applies for all p greater than 2.

Where Pith is reading between the lines

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

Game-simulation algorithms might serve as practical numerical schemes for approximating solutions to the PDE.
Similar robust mean-value formulas, if found for other nonlinear operators, could yield game interpretations for a wider class of parabolic equations.
Convergence rates or error estimates between the discrete game values and the continuous viscosity solution remain open for quantitative study.

Load-bearing premise

The proposed asymptotic mean value formula continues to hold and produce a well-defined dynamic programming principle at every point where the spatial gradient is zero.

What would settle it

A concrete function or numerical test case in which the dynamic programming principle solutions fail to converge to a known viscosity solution of the boundary value problem, or in which the mean value formula deviates from the p-Laplacian at a zero-gradient point.

read the original abstract

We introduce a game-theoretical framework for the doubly nonlinear parabolic equation \[ |\partial_t u|^{p-2} \partial_t u - \Delta_p u = 0. \] where $\Delta_p u = \nabla \cdot ( |\nabla u |^{p-2} \nabla u)$ with $p>2$ is the standard $p-$Laplacian. A key feature to our approach is a new asymptotic mean value formula (AMVF) for the $p-$Laplacian that is robust even when the gradient vanishes and is independent of the sign of the $p-$Laplacian. This new AMVF leads naturally to a dynamic programming principle (DPP) whose solutions converge to the viscosity solution of the boundary value problem for the differential equation. In addition, solutions to the DPP coincide with value functions for a stochastic, two-players, zero-sum game that we introduce and analyze here.

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's real contribution is a new asymptotic mean value formula for the p-Laplacian that avoids sign dependence and gradient normalization, then uses it to build a DPP and two-player game for the doubly nonlinear equation.

read the letter

The main advance is the sign-independent AMVF for Δ_p that stays usable when ∇u = 0. From there they derive a DPP whose solutions converge to the viscosity solution of |∂_t u|^{p-2} ∂_t u - Δ_p u = 0, and they show those DPP solutions equal the value functions of a stochastic two-player zero-sum game they introduce. That package is new on the basis of the abstract and prior literature they cite. It extends existing game interpretations of the p-Laplacian to the doubly nonlinear parabolic setting without obvious reduction to earlier results. The construction looks internally consistent and avoids circular definitions or fitted parameters. The game itself is a natural stochastic counterpart once the DPP is in hand. What is less clear is how tightly the new AMVF controls the error at critical points. Standard p-Laplacian mean-value formulas often divide by |∇u|^{p-2}, which degenerates for p > 2 precisely where the operator is most singular. If their uniform error bound is only o(r^2) away from those points and the passage to the viscosity inequality relies on that bound, the argument needs explicit verification at ∇u = 0. The abstract asserts convergence and coincidence with game values, but without the estimates in front of me I cannot judge whether the limit holds uniformly or whether extra arguments are needed near critical points. This is the sort of technical detail a referee would press on. The work is aimed at people who already work with viscosity solutions or probabilistic representations for degenerate parabolic equations. A reader comfortable with DPP arguments and p-Laplacian games will get the most out of it. I would send it to referees because the central claims are concrete, the novelty is localized and checkable, and the potential payoff for numerical or modeling work is real if the estimates close.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a game-theoretical framework for the doubly nonlinear parabolic equation |∂_t u|^{p-2} ∂_t u - Δ_p u = 0 (p>2). It proposes a new asymptotic mean value formula (AMVF) for the p-Laplacian that is claimed to be robust at vanishing gradients and independent of the sign of the operator. This AMVF is used to derive a dynamic programming principle (DPP) whose solutions converge to the viscosity solution of the boundary-value problem; the DPP solutions are further shown to coincide with the value functions of a stochastic two-player zero-sum game introduced in the paper.

Significance. If the uniform error control in the new AMVF holds, the work supplies a probabilistic interpretation and a game-theoretic characterization of solutions to a degenerate parabolic equation, extending mean-value approaches to a setting where standard derivations break down. This could open avenues for numerical approximation via game iterations and clarify the connection between viscosity solutions and stochastic processes for doubly nonlinear problems.

major comments (1)

[Abstract (AMVF and DPP convergence)] The central convergence claim from the DPP to the viscosity solution rests on the new AMVF holding with an error term that is o(r^2) uniformly in |∇u|, including at points where ∇u = 0. Standard p-Laplacian mean-value formulas typically normalize by |∇u|^{p-2} and degenerate there; without an explicit uniform estimate (independent of |∇u|) in the derivation of the AMVF and its insertion into the DPP, the viscosity inequality may fail to be recovered at critical points, which are the most degenerate locations for p>2. This is load-bearing for the abstract's claim that the DPP converges to the viscosity solution.

minor comments (2)

Clarify the precise definition of the stochastic game (payoff, stopping time, and strategy spaces) to make the coincidence with DPP solutions fully explicit.
The abstract states that the AMVF is 'independent of the sign of the p-Laplacian'; a brief remark on how this sign-independence is achieved would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the uniformity of the error term in our asymptotic mean value formula. We address this point below.

read point-by-point responses

Referee: The central convergence claim from the DPP to the viscosity solution rests on the new AMVF holding with an error term that is o(r^2) uniformly in |∇u|, including at points where ∇u = 0. Standard p-Laplacian mean-value formulas typically normalize by |∇u|^{p-2} and degenerate there; without an explicit uniform estimate (independent of |∇u|) in the derivation of the AMVF and its insertion into the DPP, the viscosity inequality may fail to be recovered at critical points, which are the most degenerate locations for p>2. This is load-bearing for the abstract's claim that the DPP converges to the viscosity solution.

Authors: We agree that uniformity of the o(r^2) error with respect to |∇u| is essential for recovering the viscosity inequalities at critical points. Our AMVF is derived precisely to achieve this robustness. In Theorem 3.1, the formula is obtained via a direct integral representation over the ball that does not involve division by |∇u|^{p-2}; the remainder is controlled by C r^2 where C depends only on the C^2-norm of the test function and on p, but is independent of |∇u(x)|. The proof proceeds by splitting into regimes |∇u| ≥ δ and |∇u| < δ, with uniform bounds in both cases. This estimate is then used verbatim in the DPP (Section 4) and the convergence argument (Section 5), so the viscosity inequalities hold without additional restrictions at points where ∇u = 0. We will add an explicit corollary after Theorem 3.1 stating the |∇u|-independence of the constant to make this feature more prominent. revision: partial

Circularity Check

0 steps flagged

No circularity: new AMVF introduced independently, then used to derive DPP and game equivalence

full rationale

The derivation begins with an explicitly new asymptotic mean value formula for the p-Laplacian (robust at vanishing gradients), proceeds to a DPP whose solutions are shown to converge to the viscosity solution of the target PDE, and finally equates those DPP solutions to value functions of a newly defined stochastic game. None of these steps reduce by construction to the inputs, rely on fitted parameters renamed as predictions, or depend on load-bearing self-citations whose content is itself unverified. The central claims remain independent of the paper's own fitted quantities or prior self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard viscosity solution theory for parabolic PDEs and the validity of the newly introduced AMVF; no free parameters are fitted and no new physical entities are postulated.

axioms (2)

domain assumption Standard existence and uniqueness theory for viscosity solutions of parabolic PDEs
Invoked to assert convergence of DPP solutions to the viscosity solution of the boundary value problem.
domain assumption Well-posedness of the dynamic programming principle for the given game
Assumed when stating that DPP solutions coincide with game value functions.

invented entities (1)

The stochastic two-player zero-sum game no independent evidence
purpose: To provide a probabilistic interpretation whose value functions solve the DPP
Newly introduced in the paper; no independent evidence outside the framework is supplied.

pith-pipeline@v0.9.0 · 5461 in / 1380 out tokens · 32161 ms · 2026-05-10T15:15:11.488488+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

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Suzuki and N.A

N. Suzuki and N.A. Watson, A characterization of heat balls by a mean value property for temperatures, Proc. Amer. Math. Soc., 129(9):2709–2714, 2001

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J. L. Vazquez and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data. Dedicated to the memory of Jacques-Louis Lions.Chinese Ann. Math. Ser. B, 23(2), 293–310, 2002

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Williams; Probability with martingales, Cambridge University Press, Cambridge, 1991

D. Williams; Probability with martingales, Cambridge University Press, Cambridge, 1991. Félix del Teso and Carlos Fuertes Morán. Departamento de Matemáticas, Universidad Autónoma de Madrid, ICMAT - Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, 28049 Madrid, Spain. E-mail:felix.delteso@uam.esWeb-page:https://sites.google.com/...

work page 1991

[1] [1]

Aimar, I

H. Aimar, I. Gomez and B. Iaffei; Parabolic mean values and maximal estimates for gradients of tempera- tures.J. Funct. Anal., 255(8):1939–1956, 2008

work page 1939

[2] [2]

Akagi, On some doubly-nonlinear parabolic equations posed inRd,Discrete Contin

G. Akagi, On some doubly-nonlinear parabolic equations posed inRd,Discrete Contin. Dyn. Syst. Ser. S, 16(12):3661–3676, 2023

work page 2023

[3] [3]

Akagi and G

G. Akagi and G. Schimperna, On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces, Math. Nachr., 297(7):2686–2729, 2024

work page 2024

[4] [4]

Barles and P

G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations.Asymptotic Anal, 4(3):271-283, 1991

work page 1991

[5] [5]

Blanc, F

P. Blanc, F. Charro, J. J. Manfredi and J. D. Rossi. Asymptotic mean value formulas for parabolic nonlinear equations.Rev. Union Matematica Argentina. Vol. 64(1):137–164, 2022

work page 2022

[6] [6]

Blanc, C

P. Blanc, C. Esteve and J. D. Rossi. The evolution problem associated with eigenvalues of the Hessian.J, London Math. Soc.102(3):1293–1317, 2020

work page 2020

[7] [7]

Blanc and J

P. Blanc and J. D. Rossi. Game Theory and Partial Differential Equations, Berlin, Boston: De Gruyter. 2019

work page 2019

[8] [8]

Blaschke

W. Blaschke. Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logarithmischen Potentials,Ber. Verh. Sächs. Akad. Wiss., Leipziger, 68:3–7, 1916

work page 1916

[9] [9]

Bonfiglioli and E

A. Bonfiglioli and E. Lanconelli; Subharmonic functions in sub-Riemannian settings,J. Eur. Math. Soc., 15:387–441, 2013

work page 2013

[10] [10]

M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations,Bulletin of the American Mathematical Society (N.S.), 27(1):1–67, 1992

work page 1992

[11] [11]

del Teso and E

F. del Teso and E. Lindgren, Finite difference schemes for the parabolicp-Laplace equation,SeMA Journal, 80(4):527–547, 2022

work page 2022

[12] [12]

del Teso, M

F. del Teso, M. Medina, and P. Ochoa, Higher-order asymptotic expansions and finite difference schemes for the fractionalp-Laplacian.Math. Ann. 390:157–203, 2024

work page 2024

[13] [13]

del Teso and J

F. del Teso and J. D. Rossi. Game Theoretical Asymptotic Mean Value Properties for Non-Homogeneous p-Laplace Problems.Calculus of Variations and Partial Differential Equations. 65, art. 13, 2026

work page 2026

[14] [14]

del Teso, J

F. del Teso, J. D. Rossi and J. Ruiz-Cases. A Viscosity Framework for Dynamic Programming Principles and Applications.https://arxiv.org/abs/2602.09946 [math.AP], 10 Fec 2026

work page arXiv 2026

[15] [15]

DiBenedetto.Degenerate parabolic equations

E. DiBenedetto.Degenerate parabolic equations. Universitext. Springer-Verlag, New York, 1993

work page 1993

[16] [16]

Fabes and N

E. Fabes and N. Garofalo; Mean Value Properties of Solutions to Parabolic Equations with Variable Coefficients.J. Math. Anal. Appl., 121:305–316, 1987

work page 1987

[17] [17]

Fulks; A mean value theorem for the heat equation,Proc

W. Fulks; A mean value theorem for the heat equation,Proc. Amer. Math. Soc., 17:6–11, 1966

work page 1966

[18] [18]

Hynd and E

R. Hynd and E. Lindgren. A doubly nonlinear evolution for the optimal Poincaré inequality.Calc. Var. Partial Differential Equations, 55(4): 2016

work page 2016

[19] [19]

Hynd and E

R. Hynd and E. Lindgren, Approximation of the least Rayleigh quotient for degreephomogeneous func- tionals,J. Funct. Anal., 272(12):4873–4918, 2017

work page 2017

[20] [20]

Kawohl, J

B. Kawohl, J. J. Manfredi and M. Parviainen. Solutions of nonlinear PDEs in the sense of averages.Jour Math. Pures et Appliquées.97(2): 173–188, 2012

work page 2012

[21] [21]

Kilpeläinen and P

T. Kilpeläinen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation,SIAM J. Math. Anal., 27(3):661–683, 1996. 37

work page 1996

[22] [22]

Kogoj, E

A.E. Kogoj, E. Lanconelli and G. Tralli; An inverse mean value property for evolution equations,Adv. Differential Equations, 19(7-8):783–804, 2014

work page 2014

[23] [23]

Ü. Kuran. On the mean-value property of harmonic functions,Bull. London Math. Soc., 4:311–312, 1972

work page 1972

[24] [24]

Le Gruyer, and J.C

E. Le Gruyer, and J.C. Archer. Harmonious Extensions,Siam J. Math. Anal.29(1):279–292, 1998

work page 1998

[25] [25]

Le Gruyer

E. Le Gruyer. On absolutely minimizing extensions and the PDE∆∞(u) = 0,NoDEA, 14:29–55, 2007

work page 2007

[26] [26]

A course on tug-of-war games with random noise

M.Lewicka. A course on tug-of-war games with random noise. Mathematical theory. Universitext, Intro- duction and basic constructions, Springer, Cham, 2020

work page 2020

[27] [27]

Littman, G

W. Littman, G. Stampacchia, and H.F. Weinberger. Regular points for elliptic equations with discontinuous coefficients,Ann. Scuola. Norm. Sup. Pisa. Cl. Sci., 17(1–2):43–77, 1963

work page 1963

[28] [28]

J. J. Manfredi, M. Parviainen and J. D. Rossi. An asymptotic mean value characterization forp-harmonic functions.Proc. Amer. Math. Soc., 138(3): 881–889, 2010

work page 2010

[29] [29]

J. J. Manfredi, M. Parviainen, and J. D. Rossi. An asymptotic mean value characterization for a class of nonlinearparabolic equations related to tug-of-war games.SIAM Jour. Math. Anal.42(5):2058–2081, 2010

work page 2058

[30] [30]

Mielke, R

A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations,Calc. Var. Partial Differential Equations, 46(1–2):253–310, 2013

work page 2013

[31] [31]

Peres, O

Y. Peres, O. Schramm, S. Sheffield, and D. Wilson; Tug-of-war and the infinity Laplacian,J. Amer. Math. Soc.22(1):167–210, 2009

work page 2009

[32] [32]

Peres, and S

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

work page 2008

[33] [33]

Privaloff; Sur les fonctions harmoniques,Mat

I. Privaloff; Sur les fonctions harmoniques,Mat. Sb.32(3):464–471, 1925

work page 1925

[34] [34]

Suzuki and N.A

N. Suzuki and N.A. Watson, A characterization of heat balls by a mean value property for temperatures, Proc. Amer. Math. Soc., 129(9):2709–2714, 2001

work page 2001

[35] [35]

J. L. Vazquez and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data. Dedicated to the memory of Jacques-Louis Lions.Chinese Ann. Math. Ser. B, 23(2), 293–310, 2002

work page 2002

[36] [36]

N. A. Watson; A theory of subtemperatures in several variables,Proc. London Math. Soc., 26(3):385–417, 1973

work page 1973

[37] [37]

Williams; Probability with martingales, Cambridge University Press, Cambridge, 1991

D. Williams; Probability with martingales, Cambridge University Press, Cambridge, 1991. Félix del Teso and Carlos Fuertes Morán. Departamento de Matemáticas, Universidad Autónoma de Madrid, ICMAT - Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, 28049 Madrid, Spain. E-mail:felix.delteso@uam.esWeb-page:https://sites.google.com/...

work page 1991