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<p>This paper establishes a probabilistic representation for the solution of the parabolic obstacle problem associated with the normalized <i>p</i>-Laplacian. We introduce a zero-sum stochastic tug-of-war game with noise in a space-time cylinder, where one player has the option to stop the game at any time to collect a payoff given by an obstacle function. We prove that the value functions of this game exist, satisfy a dynamic programming principle, and converge uniformly to the unique viscosity solution of the continuous obstacle problem as the step size <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> tends to zero.</p>

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A game-theoretic approach to the parabolic normalized p-Laplacian obstacle problem

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Hamid El Bahja

摘要

This paper establishes a probabilistic representation for the solution of the parabolic obstacle problem associated with the normalized p-Laplacian. We introduce a zero-sum stochastic tug-of-war game with noise in a space-time cylinder, where one player has the option to stop the game at any time to collect a payoff given by an obstacle function. We prove that the value functions of this game exist, satisfy a dynamic programming principle, and converge uniformly to the unique viscosity solution of the continuous obstacle problem as the step size \(\varepsilon \) ε tends to zero.