Non-local Tug-of-war and the Infinity Fractional Laplacian
نویسنده
چکیده
Motivated by the “tug-of-war” game studied in [12], we consider a “non-local” version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amount > 0 (as is done in the classical case), it is a s-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically we derive a deterministic non-local integro-differential equation that we call “infinity fractional Laplacian”. We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double obstacle problem, both problems having a natural interpretation as “tug-of-war” games.
منابع مشابه
An Existence Result for the Infinity Laplacian with Non-homogeneous Neumann Boundary Conditions Using Tug-of-war Games
In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with nonhomogeneous mixed boundary conditions. For a Lipschitz and positive function g there exists a viscosity solution of the mixed boundary value problem,
متن کاملA Mixed Problem for the Infinity Laplacian via Tug-of-war Games
In this paper we prove that a function u ∈ C(Ω) is the continuous value of the Tug-of-War game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions
متن کاملA PDE Perspective of The Normalized Infinity Laplacian
The inhomogeneous normalized infinity Laplace equation was derived from the tug-of-war game in [PSSW] with the positive right-hand-side as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized infinity Laplacian, formally written as 4∞u = | 5 u|−2 ∑n i,j=1 ∂xiu∂xju∂ xixju, is defined in a...
متن کاملNon-local Gradient Dependent Operators
In this paper we study a general class of “quasilinear non-local equations” depending on the gradient which arise from tug-of-war games. We establish a C/C/C regularity theory for these equations (the kind of regularity depending on the assumptions on the kernel), and we construct different non-local approximations of the p-Laplacian.
متن کاملBiased Tug-of-war, the Biased Infinity Laplacian, and Comparison with Exponential Cones
We prove that if U ⊂ R 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 β|∇u|+∆∞u = 0 on U that extends F , where ∆∞u = |∇u|−2 ∑ i,j uxiuxixjuxj . In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield andWilson, and de...
متن کامل