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- W1976165672 abstract "We prove that if $${Usubset mathbb {R}^n}$$ is an open domain whose closure $${overline U}$$ is compact in the path metric, and F is a Lipschitz function on ∂U, then for each $${beta in mathbb {R}}$$ there exists a unique viscosity solution to the β-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 β-biased $${epsilon}$$ -game as follows. The starting position is $${x_0 in U}$$ . At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of $${exp(betaepsilon)}$$ to 1), and the winner chooses x k with $${d(x_k,x_{k-1}) < epsilon}$$ . 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 $${u^{epsilon}(x_0)}$$ of this game exists, and that $${|u^epsilon - u|_infty to 0}$$ as $${epsilon to 0}$$ , where u is the unique extension of F to $${overline{U}}$$ that satisfies comparison with β-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 β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation." @default.
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- W1976165672 date "2009-12-10" @default.
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- W1976165672 title "Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones" @default.
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- W1976165672 doi "https://doi.org/10.1007/s00526-009-0298-2" @default.
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