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- W4301757513 abstract "We prove that if UsubsetR^n is an open domain whose closure overline{U} is compact in the path metric, and F is a Lipschitz function on partial{U}, then for each betainR there exists a unique viscosity solution to the beta-biased infinity Laplacian equation beta |nabla u| + Delta_infty u=0 on U that extends F, where Delta_infty u= |nabla u|^{-2} sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the beta-biased eps-game as follows. The starting position is x_0 in U. At the k^text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of exp(betaeps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < eps. The game ends when x_k in partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{eps}(x_0) of this game exists, and that |u^eps - u|_infty to 0 as eps to 0, where u is the unique extension of F to overline{U} that satisfies comparison with beta-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with beta-exponential cones if and only if it is a viscosity solution to the beta-biased infinity Laplacian equation." @default.
- W4301757513 created "2022-10-05" @default.
- W4301757513 creator A5020836845 @default.
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- W4301757513 date "2008-11-02" @default.
- W4301757513 modified "2023-10-16" @default.
- W4301757513 title "Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones" @default.
- W4301757513 doi "https://doi.org/10.48550/arxiv.0811.0208" @default.
- W4301757513 hasPublicationYear "2008" @default.
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