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TUG-OF-WAR AND THE INFINITY LAPLACIAN
, 2008
"... 1.1. Overview. 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 ..."
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Cited by 115 (8 self)
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1.1. Overview. 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
The infinity Laplacian, Aronsson’s equation and their generalizations
"... Abstract. The infinity Laplace equation ∆∞u = 0 arose originally as a sort of Euler–Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional ess-sup U |Du|. The more general functional ess-sup U F (x, u, Du) leads similarly to the so-called Aron ..."
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Cited by 50 (1 self)
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Abstract. The infinity Laplace equation ∆∞u = 0 arose originally as a sort of Euler–Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional ess-sup U |Du|. The more general functional ess-sup U F (x, u, Du) leads similarly to the so-called Aronsson equation AF [u] = 0. In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to L ∞ variational problems, including two-person game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional L ∞ variational problems. 1.
A deterministic-control-based approach to motion by curvature
- Comm. Pure Appl. Math
"... by curvature ..."
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An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions
, 2009
"... We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls. ..."
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Cited by 33 (6 self)
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We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls.
On the definition and properties of p-harmonious functions
, 2009
"... We consider functions that satisfy the identity uε(x) = ..."
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Cited by 24 (11 self)
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We consider functions that satisfy the identity uε(x) =
A finite difference approach to the infinity Laplace equation and tug-of-war games
- TRANS. AMER. MATH. SOC
, 2009
"... We present a modified version of the two-player “tug-of-war” game introduced by Peres, Schramm, Sheffield, and Wilson [18]. This new tug-ofwar game is identical to the original except near the boundary of the domain ∂Ω, but its associated value functions are more regular. The dynamic programming pri ..."
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Cited by 22 (6 self)
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We present a modified version of the two-player “tug-of-war” game introduced by Peres, Schramm, Sheffield, and Wilson [18]. This new tug-ofwar game is identical to the original except near the boundary of the domain ∂Ω, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results. We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tug-of-war players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff function is nonnegative. We also show that uniqueness holds in certain cases for sign-changing running payoff functions which are sufficiently small. In the limit ε → 0, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation. We also obtain several new results for the normalized infinity Laplace equation −∆∞u = f. In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous f, and continuous boundary data, as well as the uniqueness of solutions to this problem in the generic case. We present a new elementary proof of uniqueness in the case that f> 0, f < 0, or f ≡ 0. The stability of the solutions with respect to f is also studied, and an explicit continuous dependence estimate from f ≡ 0 is obtained.
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO Tug-of-war Games
"... We characterize solutions to the homogeneous parabolic p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for t ..."
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Cited by 18 (9 self)
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We characterize solutions to the homogeneous parabolic p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for these game approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.
Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher
- SIAM J. Numer. Anal
"... Abstract. The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case st ..."
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Cited by 17 (4 self)
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Abstract. The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the Monge-Ampère equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton’s method. We prove convergence of Newton’s method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to non-differentiable. 1.
Dynamic programming principle for tug-of- war games with noise,
- ESAIM: Control, Optimisation and Calculus of Variations, COCV,
, 2012
"... Abstract. Consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game posi ..."
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Cited by 15 (8 self)
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Abstract. Consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F . We give a detailed proof of the fact that the value functions of this game satisfy the dynamic programming principle for x ∈ Ω with u(y) = F (y) when y ̸ ∈ Ω. This principle implies the existence of quasioptimal Markovian strategies.
BIASED TUG-OF-WAR, THE BIASED INFINITY LAPLACIAN, AND COMPARISON WITH EXPONENTIAL CONES
, 811
"... 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 ..."
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Cited by 14 (1 self)
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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.
