Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian

Abstract

Abstract. 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.

Perron’s Method and Wiener’s Theorem for a Nonlocal Equation

“…In addition, in the papers [4], [3] and [9] different types of non-local nonlinear equations are introduced.…”

“…where B R \ B r denotes the ring domain r < |x| < R. In order to use ω as a barrier 4 , we need to assure that it attains its boundary values.…”

On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian

“…For u ∈ L p−1 (R N ), x ∈ R N and ε > 0, we let (−∆) π N 2 Γ(1 − s) and Γ is the usual Gamma function (see, e.g., [5,8,9,10,11] for the linear case p = 2, and [24,25] for the general case p ∈ (1, ∞)). The fractional p-Laplacian (−∆) provided that the limit exists.…”

Some integrability estimates for solutions of the fractional p-Laplace equation

“…It is a generator of a strongly continuous contractive semigroup on L 2 (R n ) that can be extended to contraction semigroup on L p (R n ) for p ∈ [1, ∞] ( [3,8]). The Dirichlet boundary problem of L α,2 has been intensively investigated and many fundamental results have been proved, we refer the reader to [2,4,8,12,14] and the references therein for a fuller treatment of this topic. As a nonlinear generalization of L α,2 , L α,p has been extensively explored in recent years ( [1,5,10]).…”