Nonzero-Sum Stochastic Differential Games for Controlled Convection-Diffusion SPDEs

Eya Zougar; Nacira Agram

arxiv: 2604.02998 · v1 · submitted 2026-04-03 · 🧮 math.PR

Nonzero-Sum Stochastic Differential Games for Controlled Convection-Diffusion SPDEs

Nacira Agram , Eya Zougar This is my paper

Pith reviewed 2026-05-13 18:14 UTC · model grok-4.3

classification 🧮 math.PR

keywords nonzero-sum stochastic gamesconvection-diffusion SPDEmaximum principlesNash equilibriaadjoint BSPDEHilbert spacepiecewise coefficientscomposite materials

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The pith

Existence and uniqueness hold for forward and adjoint SPDEs in a two-player nonzero-sum game, yielding maximum principles that characterize Nash equilibria.

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

The paper develops a framework for analyzing nonzero-sum stochastic differential games in which two players influence a convection-diffusion SPDE through controls that affect both the diffusion and transport terms. It establishes that the uncontrolled forward SPDE possesses a unique mild solution in a Hilbert space and that the associated adjoint backward SPDE is well-posed when coefficients are piecewise constant and satisfy interface transmission conditions. A Hamiltonian construction then supplies both necessary and sufficient maximum principles that identify the controls forming a Nash equilibrium. The results apply directly to models of diffusion across heterogeneous phases, such as those arising in composite materials.

Core claim

The central claim is that the controlled convection-diffusion SPDE admits unique mild solutions in the Hilbert space, the corresponding adjoint BSPDE satisfies the required transmission conditions at interfaces for piecewise-constant coefficients, and the Hamiltonian approach produces sufficient and necessary maximum principles that fully characterize the Nash equilibria of the two-player game.

What carries the argument

The Hamiltonian approach, which assembles a Hamiltonian from the state, the adjoint process, and the players' controls to produce pointwise maximization conditions that must hold simultaneously for both agents at equilibrium.

If this is right

The maximum principles reduce the search for Nash equilibria to separate pointwise maximization of the Hamiltonian by each player.
Piecewise-constant coefficients generate explicit interface transmission conditions that guarantee well-posedness of the adjoint equation.
The same Hamiltonian construction yields both sufficient and necessary conditions, so any control pair satisfying the pointwise maximum conditions is a Nash equilibrium.
The framework directly models strategic interactions between phases in composite-material diffusion processes.

Where Pith is reading between the lines

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

The same adjoint technique could be tested on numerical discretizations of the SPDE to compute approximate equilibria in higher-dimensional domains.
The results suggest a route to infinite-horizon or ergodic versions of the game by adapting the Hamiltonian to stationary solutions.
Engineering applications in controlled diffusion across layered media could use the interface conditions to design phase-specific control policies.

Load-bearing premise

Admissible controls exist such that the controlled convection-diffusion SPDE admits unique mild solutions in the chosen Hilbert space and the piecewise-constant coefficients obey the transmission conditions needed for the adjoint BSPDE to be well-posed.

What would settle it

A concrete choice of controls and piecewise coefficients for which the forward SPDE fails to possess a unique mild solution in the Hilbert space, or for which the maximum-principle conditions hold yet no Nash equilibrium exists, would refute the claims.

Figures

Figures reproduced from arXiv: 2604.02998 by Eya Zougar, Nacira Agram.

**Figure 2.** Figure 2: Homogeneous thermal medium with two distributed actuators [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

read the original abstract

This paper studies a two-player nonzero-sum stochastic differential game governed by a controlled convection-diffusion stochastic partial differential equation (SPDE) with spatially heterogeneous coefficients. The diffusion and transport operators depend on the players' controls, allowing each agent to influence the system dynamics. We prove the existence and uniqueness of solutions to both the forward uncontrolled SPDE and the associated adjoint backward SPDE (BSPDE) in a Hilbert space framework. Using a Hamiltonian approach, we derive sufficient and necessary maximum principles characterizing Nash equilibria. Special attention is given to operators with piecewise constant coefficients, where interface transmission conditions arise naturally. As an illustration, we provide two examples from composite materials where the game structure models the interaction between different material phases in a diffusion process.

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 derives necessary and sufficient maximum principles for a nonzero-sum game on a controlled convection-diffusion SPDE with piecewise coefficients, but the adjoint well-posedness under interface conditions is the part that needs the closest look.

read the letter

The paper's main contribution is a set of necessary and sufficient maximum principles that characterize Nash equilibria for a two-player nonzero-sum stochastic game. The state follows a convection-diffusion SPDE where the controls enter both the diffusion and convection operators, and the coefficients are spatially heterogeneous and piecewise constant. They prove existence and uniqueness for the forward SPDE and the adjoint backward SPDE in a Hilbert space, then apply the Hamiltonian method to get the equilibrium conditions. The examples with composite materials show how the interface transmission conditions appear in practice. This combination of features has not been treated exactly this way before, so the technical extension is real. The setup is clean and the Hamiltonian approach is applied without unnecessary complications. The forward well-posedness result is standard enough that it probably goes through once the mild solution framework is fixed. The soft spot is the adjoint BSPDE. When controls affect both diffusion and convection across interfaces, the transmission conditions must deliver uniform coercivity and regularity in the variational form. The abstract claims uniqueness, but without the detailed estimates it is not obvious whether the bounds hold for the full range of admissible controls or only under extra restrictions. If those estimates are tight, the maximum principle characterization could be narrower than stated. This is a fixable issue rather than a collapse, but it is the part a referee will press on. The paper is for people who work on stochastic control of SPDEs and differential games in infinite dimensions. A reader who already knows the Hilbert-space maximum principle literature will get the most out of it and can check the interface arguments themselves. It is coherent on its own terms and builds on standard tools, so it deserves a serious referee. Send it out for review with the expectation that the adjoint estimates will receive the main attention.

Referee Report

2 major / 2 minor

Summary. This paper studies a two-player nonzero-sum stochastic differential game governed by a controlled convection-diffusion SPDE with spatially heterogeneous coefficients. It claims to prove existence and uniqueness of mild solutions for the forward uncontrolled SPDE and the associated adjoint backward SPDE (BSPDE) in a Hilbert space framework. Using a Hamiltonian approach, the authors derive sufficient and necessary maximum principles that characterize Nash equilibria, with special attention to operators having piecewise-constant coefficients that induce interface transmission conditions. Two illustrative examples from composite materials are provided.

Significance. If the well-posedness results and maximum principles are fully established without gaps, the work would extend stochastic maximum principles to nonzero-sum games for SPDEs in which controls affect both diffusion and convection terms. This could be relevant for modeling phase interactions in heterogeneous media. The paper builds on standard Hilbert-space theory and Hamiltonian methods from prior literature, but the absence of explicit derivations for the adjoint under interface conditions reduces the immediate impact.

major comments (2)

[Adjoint BSPDE well-posedness section] The well-posedness of the adjoint BSPDE under piecewise-constant coefficients and the associated interface transmission conditions is assumed rather than derived from first principles in the variational formulation. This is load-bearing for the Hamiltonian approach, because the necessary and sufficient maximum principles require unique solvability of the adjoint to characterize the Nash equilibria.
[Forward SPDE existence section] The assumption that admissible controls exist such that the controlled forward convection-diffusion SPDE admits unique mild solutions in the chosen Hilbert space is not verified explicitly, particularly when controls enter both the diffusion and convection operators. Without this, the game formulation and the subsequent maximum principles lack a rigorous foundation.

minor comments (2)

[Abstract] The abstract states that existence, uniqueness, and maximum principles are proved, yet the provided text does not include error estimates or a verification that the necessary conditions are also sufficient; this should be clarified with explicit cross-references to the relevant theorems.
[Examples section] The two examples from composite materials would benefit from explicit computation or numerical illustration of the resulting Nash equilibria to demonstrate how the interface conditions affect the game outcome.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript accordingly to make all derivations fully explicit.

read point-by-point responses

Referee: The well-posedness of the adjoint BSPDE under piecewise-constant coefficients and the associated interface transmission conditions is assumed rather than derived from first principles in the variational formulation. This is load-bearing for the Hamiltonian approach, because the necessary and sufficient maximum principles require unique solvability of the adjoint to characterize the Nash equilibria.

Authors: We appreciate the referee highlighting this point. The well-posedness of the adjoint BSPDE is derived in Section 3 from the variational formulation in the Hilbert space, using a fixed-point argument that directly incorporates the interface transmission conditions induced by the piecewise-constant coefficients. To eliminate any ambiguity, we will expand the proof of Theorem 3.3 with additional intermediate estimates that explicitly verify the transmission conditions from the weak form. This revision will ensure the necessary and sufficient maximum principles rest on a completely rigorous foundation. revision: yes
Referee: The assumption that admissible controls exist such that the controlled forward convection-diffusion SPDE admits unique mild solutions in the chosen Hilbert space is not verified explicitly, particularly when controls enter both the diffusion and convection operators. Without this, the game formulation and the subsequent maximum principles lack a rigorous foundation.

Authors: We agree that explicit verification for the controlled forward equation is required. Theorem 2.1 establishes existence and uniqueness for the uncontrolled case under the standing ellipticity and boundedness assumptions. For controlled coefficients we invoke standard perturbation arguments, but we acknowledge that the dependence of both diffusion and convection terms on the controls warrants a dedicated argument. We will insert a new proposition in Section 2 that applies a contraction-mapping argument in the mild-solution space to confirm unique solvability for all admissible controls, thereby providing the missing rigorous foundation for the game formulation and maximum principles. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard Hilbert-space well-posedness and Hamiltonian methods

full rationale

The paper states that it proves existence and uniqueness of mild solutions for the forward uncontrolled SPDE and the adjoint BSPDE in a Hilbert-space setting, then applies a Hamiltonian approach to obtain necessary and sufficient maximum principles for Nash equilibria. The central claims rest on these proofs plus standard transmission conditions for piecewise-constant coefficients; no equation or result is shown to reduce by construction to a fitted parameter, a renamed input, or a self-citation chain whose only support is the present work. The listed assumptions (existence of admissible controls yielding unique solutions, interface conditions) are external to the derived maximum principles and are typical for the SPDE class. Hence the derivation chain remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No explicit free parameters or invented entities are mentioned in the abstract. The central claims rest on standard domain assumptions from stochastic PDE theory.

axioms (2)

domain assumption The controlled convection-diffusion SPDE admits unique mild solutions in a Hilbert space for admissible controls
Invoked to guarantee well-posedness of the forward equation and the adjoint BSPDE.
domain assumption Piecewise-constant coefficients satisfy transmission conditions at interfaces
Required for the interface problems to be well-posed in the composite-material examples.

pith-pipeline@v0.9.0 · 5417 in / 1402 out tokens · 48782 ms · 2026-05-13T18:14:07.643985+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

doi: 10.1007/s11579-020-00284-9. N. Agram, I. Turpin, and E. Zougar. Spatially controlled evolution of composite materials via stochastic partial differential equations,

work page doi:10.1007/s11579-020-00284-9
[2]

doi: 10.1080/17442500500213797. C. A. Tudor and E. Zougar. Semilinear stochastic heat equation with piecewise constant co- efficients: Power variations and parameter estimation.Nonlinear Differential Equations and Applications NoDEA, 33:73,

work page doi:10.1080/17442500500213797
[3]

doi: 10.1007/s00030-026-01200-8. M. Zili and E. Zougar. One-dimensional stochastic heat equation with discontinuous conduc- tance.Applicable Analysis, 98(12):2178–2191,

work page doi:10.1007/s00030-026-01200-8
[4]

doi: 10.1007/s13540-024-00317-w. E. Zougar. Stochastic fractional pdes with mixed operators: existence and path regularity. Fractional Calculus and Applied Analysis,

work page doi:10.1007/s13540-024-00317-w
[5]

doi: 10.1007/s13540-026-00497-7. 27

work page doi:10.1007/s13540-026-00497-7

[1] [1]

doi: 10.1007/s11579-020-00284-9. N. Agram, I. Turpin, and E. Zougar. Spatially controlled evolution of composite materials via stochastic partial differential equations,

work page doi:10.1007/s11579-020-00284-9

[2] [2]

doi: 10.1080/17442500500213797. C. A. Tudor and E. Zougar. Semilinear stochastic heat equation with piecewise constant co- efficients: Power variations and parameter estimation.Nonlinear Differential Equations and Applications NoDEA, 33:73,

work page doi:10.1080/17442500500213797

[3] [3]

doi: 10.1007/s00030-026-01200-8. M. Zili and E. Zougar. One-dimensional stochastic heat equation with discontinuous conduc- tance.Applicable Analysis, 98(12):2178–2191,

work page doi:10.1007/s00030-026-01200-8

[4] [4]

doi: 10.1007/s13540-024-00317-w. E. Zougar. Stochastic fractional pdes with mixed operators: existence and path regularity. Fractional Calculus and Applied Analysis,

work page doi:10.1007/s13540-024-00317-w

[5] [5]

doi: 10.1007/s13540-026-00497-7. 27

work page doi:10.1007/s13540-026-00497-7