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

In this paper, we study the parabolic inhomogeneous beta-biased infinity Laplacian equation arising from the beta-biased tug-of-war u(t) - Delta(beta)(infinity)u = f(x, t), where beta is a fixed constant and Delta(beta)(infinity) is the beta-biased infinity Laplacian operator Delta(beta)(infinity)u = Delta(N)(infinity)u + beta vertical bar Du vertical bar related to the game theory named beta-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, we show some explicit solutions.