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BIASED TUG-OF-WAR, THE BIASED INFINITY LAPLACIAN, AND COMPARISON WITH EXPONENTIAL CONES (811)
| Citations: | 14 - 1 self |
BibTeX
@MISC{Peres811biasedtug-of-war,,
author = {Yuval Peres and Gábor Pete and Stephanie Somersille},
title = {BIASED TUG-OF-WAR, THE BIASED INFINITY LAPLACIAN, AND COMPARISON WITH EXPONENTIAL CONES},
year = {811}
}
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Abstract
Abstract. We prove that if U ⊂ R n is an open domain whose closure U is compact in the path metric, and F is a Lipschitz function on ∂U, then for each β ∈ R there exists a unique viscosity solution to the β-biased infinity Laplacian equation on U that extends F, where ∆∞u = |∇u| β|∇u | + ∆∞u = 0 −2 i,j uxiuxixjuxj. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ǫ-game as follows. The starting position is x0 ∈ U. At the k th step the two players toss a suitably biased coin (in our key example, player I wins with odds of exp(βǫ) to 1), and the winner chooses xk with d(xk, xk−1) < ǫ. The game ends when xk ∈ ∂U, and player II pays the amount F(xk) to player I. We prove that the value u ǫ (x0) of this game exists, and that ‖u ǫ − u‖ ∞ → 0 as ǫ → 0, where u is the unique extension of F to 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. 1.
Keyphrases
comparison exponential cone biased infinity laplacian exponential cone biased tug-of-war infinity laplacian equation key example linear cone unique extension continuous function satisfies th step open domain tug-of-war idea starting position unique viscosity solution viscosity solution biased game player ii lipschitz function
