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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Tug-of-war and the infinity Laplacian
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by Yuval Peres, Oded Schramm, Scott Sheffield and David B. Wilson HTML | PDF
J. Amer. Math. Soc. 22 (2009), 167-210

Abstract:

We prove that every bounded Lipschitz function $F$ on a subset $Y$ of a length space $X$ admits a tautest extension to $X$, i.e., a unique Lipschitz extension $u:X \rightarrow \mathbb {R}$ for which $\operatorname {Lip}_U u =\operatorname {Lip}_{\partial U} u$ for all open $U \subset X\smallsetminus Y$. This was previously known only for bounded domains in $\mathbb {R}^n$, in which case $u$ is infinity harmonic; that is, a viscosity solution to $\Delta _\infty u = 0$, where \[ \Delta _\infty u = |\nabla u|^{-2} \sum _{i,j} u_{x_i} u_{x_ix_j} u_{x_j}.\] We also prove the first general uniqueness results for $\Delta _{\infty } u = g$ on bounded subsets of $\mathbb {R}^n$ (when $g$ is uniformly continuous and bounded away from $0$) and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of $u$. Let $u^\varepsilon (x)$ be the value of the following two-player zero-sum game, called tug-of-war: fix $x_0=x\in X \smallsetminus Y$. At the $k^{\mathrm {th}}$ turn, the players toss a coin and the winner chooses an $x_k$ with $d(x_k, x_{k-1})< \varepsilon$. The game ends when $x_k \in Y$, and player I’s payoff is $F(x_k) - \frac {\varepsilon ^2}{2}\sum _{i=0}^{k-1} g(x_i)$. We show that $\|u^\varepsilon - u\|_{\infty } \to 0$. Even for bounded domains in $\mathbb {R}^n$, the game theoretic description of infinity harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity harmonic functions in the unit disk with boundary values supported in a $\delta$-neighborhood of a Cantor set on the unit circle.
References
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Additional Information
  • Yuval Peres
  • Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052, and Department of Statistics, 367 Evans Hall, University of California, Berkeley, California 94720
  • MR Author ID: 137920
  • Oded Schramm
  • Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052
  • Scott Sheffield
  • Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052, and Department of Statistics, 367 Evans Hall, University of California, Berkeley, California 94720
  • Address at time of publication: Courant Institute, 251 Mercer Street, New York, New York 10012
  • David B. Wilson
  • Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052
  • Received by editor(s): July 11, 2006
  • Published electronically: July 28, 2008
  • Additional Notes: Research of the first and third authors was supported in part by NSF grants DMS-0244479 and DMS-0104073.
  • © Copyright 2008 by the authors. This paper or any part thereof may be reproduced for non-commercial purposes.
  • Journal: J. Amer. Math. Soc. 22 (2009), 167-210
  • MSC (2000): Primary 91A15, 91A24, 35J70, 54E35, 49N70
  • DOI: https://doi.org/10.1090/S0894-0347-08-00606-1
  • MathSciNet review: 2449057