Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

Abstract

Abstract In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war u t - Δ ∞ β ⁢ u = f ⁢ ( x , t ) , {u_{t}}-\Delta_{\infty}^{\beta}u=f(x,t), where β is a fixed constant and Δ ∞ β {\Delta_{\infty}^{\beta}} is the β-biased infinity Laplacian operator Δ ∞ β ⁢ u = Δ ∞ N ⁢ u + β ⁢ | D ⁢ u | \Delta_{\infty}^{\beta}u=\Delta_{\infty}^{N}u+\beta\lvert Du\rvert related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when f = 0 {f=0} , we show some explicit solutions.