A stochastic differential game for the inhomogeneous \(\infty \)-Laplace equation. (English) Zbl 1192.91025

The authors study a zero-sum stochastic differential game with terminal payoff \(h\in C(\overline G,\mathbb{R}\setminus\{0\})\) and running payoff \(q\in C(\partial G,\mathbb{R})\), where \(G\subset\mathbb{R}^m\) is a bounded \(C^2\) domain. The game is played until the state process exits the domain. The main result establishes a characterization of the game value as the unique viscosity solution \(u\) of the equation \(-2\Delta_\infty u= h\) in \(G\) with boundary data \(q\).


91A15 Stochastic games, stochastic differential games
91A23 Differential games (aspects of game theory)
35J70 Degenerate elliptic equations
49L20 Dynamic programming in optimal control and differential games
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[1] Aronsson, G. (1967). Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6 551-561. · Zbl 0158.05001
[2] Aronsson, G. (1972). A mathematical model in sand mechanics: Presentation and analysis. SIAM J. Appl. Math. 22 437-458. JSTOR: · Zbl 0242.34048
[3] Atar, R. and Budhiraja, A. (2008). On near optimal trajectories for a game associated with the \infty -Laplacian. Trans. Amer. Math. Soc. 360 77-101. · Zbl 1233.91026
[4] Barron, E. N., Evans, L. C. and Jensen, R. (2009). The infinity Laplacian, Aronsson’s equation and their generalizations. · Zbl 1125.35019
[5] Crandall, M. G., Kocan, M., Lions, P. L. and Swiech, A. (1999). Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations. Electron. J. Differential Equations 24. · Zbl 0927.35029
[6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049
[7] Fleming, W. H. and Souganidis, P. E. (1989). On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 293-314. · Zbl 0686.90049
[8] Jensen, R. (1993). Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123 51-74. · Zbl 0789.35008
[9] Kohn, R. V. and Serfaty, S. (2006). A deterministic-control-based approach to motion by curvature. Comm. Pure Appl. Math. 59 344-407. · Zbl 1206.53072
[10] Peres, Y., Schramm, O., Sheffield, S. and Wilson, D. B. (2009). Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc. 22 167-210. · Zbl 1206.91002
[11] Soner, H. M. and Touzi, N. (2003). A stochastic representation for mean curvature type geometric flows. Ann. Probab. 31 1145-1165. · Zbl 1080.60076
[12] Swiech, A. (1996). Another approach to the existence of value functions of stochastic differential games. J. Math. Anal. Appl. 204 884-897. · Zbl 0870.90107
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