Tug-of-war and the infinity Laplacian
@article{Peres2006TugofwarAT, title={Tug-of-war and the infinity Laplacian}, author={Yuval Peres and Oded Schramm and Scott Sheffield and David Bruce Wilson}, journal={Journal of the American Mathematical Society}, year={2006}, volume={22}, pages={167-210} }
We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a unique absolutely minimal (AM) extension to X, i.e., a unique Lipschitz extension u : X → ℝ for which Lip U u = Lip ∂u u for all open U ⊂ X \ Y.
368 Citations
On the local and global regularity of tug-of-war games
- Mathematics
- 2018
We study a two player zero-sum tug-of-war game with varying probabilities that depend on the game location x. In particular, we show that the value of the game is locally asymptotically Hölder…
Tug-of-war games with varying probabilities and the normalized p ( x )-laplacian
- Mathematics
- 2017
We study a two player zero-sum tug-of-war game with varying probabilities that depend on the game location x. In particular, we show that the value of the game is locally asymptotically Holder…
Tug-of-War and Infinity Laplace Equation with Vanishing Neumann Boundary Condition
- Mathematics
- 2011
We study a version of the stochastic “tug-of-war” game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an…
Gradient and Lipschitz Estimates for Tug-of-War Type Games
- MathematicsSIAM J. Math. Anal.
- 2021
A random step size tug-of-war game is defined, and it is shown that the gradient of a value function exists everywhere, and an improved Lipschitz estimate is established when boundary values are close to a plane.
Regularity properties of tug-of-war games and normalized equations
- Mathematics
- 2017
We prove local Lipschitz continuity and Harnack’s inequality for value functions of the stochastic game tug-of-war with noise and running payo . As a consequence, we obtain game-theoretic proofs for…
Tug-of-War games and the infinity Laplacian with spatial dependence
- Mathematics
- 2013
In this paper we look for PDEs that arise as limits of values of
tug-of-war games when the possible movements of the game are taken
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Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities
- MathematicsPotential Analysis
- 2019
We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in Ω ⊂ ℝn. The method of the proof is based on a game-theoretic idea to estimate…
Tug-of-War games and parabolic problems with spatial and time dependence
- Mathematics
- 2012
In this paper we use probabilistic arguments (Tug-of-War games) to obtain existence of viscosity solutions to a parabolic problem of the form $$ {cases} K_{(x,t)}(D u)u_t (x,t)= \frac12 $.
An obstacle problem for Tug-of-War games
- Mathematics
- 2013
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic…
A P ] 2 8 A pr 2 00 6 Tug-of-war and the infinity Laplacian
- Mathematics
- 2005
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 → R for which LipUu = Lip∂Uu for all open…
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