Tug-of-War and Infinity Laplace Equation with Vanishing Neumann Boundary Condition

Abstract

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 appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Laplace equation with vanishing Neumann boundary condition.2010 Mathematics Subject Classification. Primary 35J70, 91A15, 91A24.

General existence of solutions to dynamic programming equations

“…Recently, the deterministic or stochastic game-theoretic approach to various nonlinear partial differential equations has attracted a lot of attention; see, for example, a variety of games in [12,13,22,23,9,4,21,1,7] and related topics in [15,20,19,24]. The results in the literature so far can be summarized in the following way.…”

Game theoretical methods in PDEs

“…In the case p = 1 the game is naturally related to the mean curvature flow [8] and functions of least gradient. Other extensions include the obstacle problems [15], finite difference schemes [2], equations with right hand side f = 0, mixed boundary data [1,3] and parabolic equations [9,14].…”

“…It is well known that the viscosity solutions to u = 0 coincide with the classical solutions. An avantage of working with the above, seemingly, more complex notion, is that the limiting properties of u follow quite naturally from the mean value property (1). Namely, replacing the increments u (x ± e i ) − u (x) in the discontinuous u in (1), by the same increments in the smooth φ, applying Taylor's expansion and taking into account the assumed sign of u − φ, yields the sign of φ, wheras the first derivatives cancel out due to the symmetry in (1).…”

The infinity Laplacian with a transport term

“…We also quote the recent Refs. [20][21][22][23] related with the interplay between tug-of-war games and the infinity Laplacian.…”