An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games

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].…”

Waiting time effect for motion by positive second derivatives and applications

“…This approximation gives an interesting representation of the solutions to the equation. See also [17] for the deterministic game approach to general elliptic and parabolic equations and [22][23][24][25] for a stochastic tug-of-war game approach to the p-Laplace equation with p > 1. Related extensions of this new method to the Heisenberg group are recently addressed in [9,10].…”

Modica type gradient estimates for an inhomogeneous variant of the normalized p-Laplacian evolution

“…Recently, there has been increasing attention about the equation of the so-called normalized p-Laplacian evolution |Du| 2−p div(|Du| p−2 Du) = u t , 1 < p < ∞, (1.1) see [2,11,24,20,3,21]. Eq.…”

“…(1.1) is an evolution associated with the p-Laplacian that interpolates between the motion by mean curvature, which corresponds to the case p = 1, and the heat equation, corresponding to p = 2. In the interesting paper [24] solutions to (1.1) have been characterized by asymptotic mean value properties. These properties are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded.…”