A game-theoretical interpretation for a doubly nonlinear parabolic equation

1. Problem Statement

The paper addresses the doubly nonlinear parabolic equation:
$$|\partial_t u|^{p-2} \partial_t u - \Delta_p u = 0$$
where $p > 2$ and $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$ is the standard $p$-Laplacian.

Key Characteristics of the Equation:

• Double Nonlinearity: The nonlinearity appears in both the time derivative (via the term $|\partial_t u|^{p-2} \partial_t u$) and the spatial diffusion (via the $p$-Laplacian).
• Homogeneity: The equation is $(p-1)$-homogeneous.
• Settings: The authors analyze the problem in two settings:
  1. Bounded Domains: The Dirichlet problem with initial and boundary data.
  2. Whole Space ($\mathbb{R}^d$): The Cauchy problem with initial data.

The primary goal is to provide a game-theoretical interpretation for this equation, characterizing its viscosity solutions as limits of value functions from a specific stochastic, two-player, zero-sum game.

2. Methodology

The authors employ a multi-step methodology bridging asymptotic analysis, dynamic programming, and stochastic game theory.

A. New Asymptotic Mean Value Formula (AMVF)

The core technical innovation is the derivation of a new AMVF for the $p$-Laplacian tailored for game theory.

• Robustness: Unlike previous formulas (e.g., in [13]), this new AMVF is independent of the sign of the $p$-Laplacian and remains valid even when the gradient vanishes ($\nabla u = 0$).
• Derivation: It utilizes a numerical identity for geometric means: $a^\alpha b^{1-\alpha} = \inf_{c>0} \{ \alpha c^{1-\alpha} a + (1-\alpha) c^{-\alpha} b \}$.
• Operator Definition: The averaging operator $A_\epsilon[\phi](x)$ is defined as a combination of suprema, infima, and averages over balls with radii depending on a parameter $c \in [m(\epsilon), M(\epsilon)]$.
  $$A_\epsilon[\phi](x) = \frac{1}{2} \max \left[ \inf_{c} (\dots), \phi(x) \right] + \frac{1}{2} \min \left[ \sup_{c} (\dots), \phi(x) \right]$$
• Asymptotic Expansion: For smooth functions, the operator satisfies:
  $$A_\epsilon[\phi](x) = \phi(x) + \frac{\epsilon^2}{2} (\Delta_p \phi(x))^{\frac{1}{p-1}} + o(\epsilon^2)$$
  For the parabolic case, it relates to the time derivative:
  $$A_\epsilon[\phi](x, t) = \phi(x, t + \epsilon^2/2) + \frac{\epsilon^2}{2} \left( (\Delta_p \phi)^{\frac{1}{p-1}} - \partial_t \phi \right) + o(\epsilon^2)$$

B. Dynamic Programming Principle (DPP)

Using the AMVF, the authors formulate a discrete-time Dynamic Programming Principle (DPP) that acts as an explicit Euler scheme for the PDE:
$$u_\epsilon(x, t) = A_\epsilon[u_\epsilon](x, t - \epsilon^2/2)$$
This DPP is analyzed for existence, uniqueness, and stability (comparison principles) in both bounded domains and the whole space.

C. Stochastic Game Construction

The authors construct a two-player zero-sum game (Player I maximizes, Player II minimizes) that corresponds to the DPP.

• Mechanics: At each step, a fair coin toss determines which player initiates a move.
• Move Options:
  1. Time Update: A player can choose to stay in space but decrease time by $\epsilon^2/2$.
  2. Spatial Move: If a move is initiated, the opponent selects a parameter $c$. Then, a biased coin determines the outcome:
     • Heads: The current player chooses the next spatial position within a ball of radius $\epsilon^2 c^{1-\alpha}$.
     • Tails: A "Tug-of-War with noise" is played within a ball of radius $\gamma \epsilon c^{-\alpha/2}$.
• Termination: The game ends when the spatial position exits the domain or time runs out ($t \le 0$). Payoffs are determined by boundary data $g$ or initial data $u_0$.

3. Key Contributions

1. Sign-Independent AMVF: The introduction of an AMVF for the $p$-Laplacian that does not require prior knowledge of the sign of $\Delta_p u$ and handles the critical case $\nabla u = 0$. This is crucial for parabolic problems where the sign of the Laplacian depends on the unknown time derivative.
2. Characterization of Viscosity Solutions: Proving that viscosity solutions to the doubly nonlinear parabolic equation are exactly the limits of the solutions to the associated AMVF/DPP as $\epsilon \to 0$.
3. Game-Theoretical Value: Establishing that the game described above has a value, and this value coincides exactly with the unique solution to the DPP.
4. Convergence Results:
   • Bounded Domains: Proving that the DPP solutions converge uniformly to the unique viscosity solution of the Dirichlet problem.
   • Whole Space: Proving convergence (along subsequences generally, and for the whole family under exponential decay assumptions of initial data) to a viscosity solution of the Cauchy problem.

4. Main Results

• Theorem 2.4 & 2.6 (Asymptotic Expansions): Rigorous error estimates for the AMVF operator $A_\epsilon$ applied to $C^2$ and $C^3$ functions, including cases where the gradient vanishes.
• Theorem 2.8 (Bounded Domain Convergence): For the Dirichlet problem, the unique solution $u_\epsilon$ of the DPP converges locally uniformly to the unique viscosity solution $u$ of the PDE as $\epsilon \to 0$.
• Theorem 2.9 (Game Value): The game defined by the rules (I)-(III) has a value $u_\epsilon$, which is the unique solution to the DPP.
• Theorem 2.10 & 6.9 (Whole Space Convergence):
  • Existence of a convergent subsequence of DPP solutions to a viscosity solution in $\mathbb{R}^d$.
  • If the initial data $u_0$ satisfies an exponential decay condition ($|u_0(x)| \le C e^{-|x|}$), the entire family $u_\epsilon$ converges uniformly to the viscosity solution.
• Comparison Principles: The paper establishes comparison principles for the DPP, which are essential for proving uniqueness and stability.

5. Significance

• Bridging Analysis and Probability: This work extends the well-known connection between the $p$-Laplacian and Tug-of-War games (previously established for elliptic and normalized parabolic cases) to the more complex doubly nonlinear setting.
• Handling Singularities: The robustness of the new AMVF at points where $\nabla u = 0$ is a significant technical advancement, as these points are traditionally difficult to handle in $p$-Laplacian analysis.
• Numerical Implications: The DPP formulation provides a natural explicit Euler scheme for solving these doubly nonlinear equations numerically. The game interpretation offers a probabilistic Monte Carlo method for approximating solutions.
• Theoretical Framework: It provides a unified framework for understanding doubly nonlinear parabolic equations through the lens of stochastic games, offering new tools for proving existence, uniqueness, and regularity properties via probabilistic arguments.

In summary, the paper successfully constructs a stochastic game whose value function converges to the viscosity solution of a challenging doubly nonlinear parabolic PDE, overcoming technical hurdles related to gradient vanishing and sign dependence through a novel asymptotic mean value formula.