A weighted eigenvalue problem of the biased infinity Laplacian*

We study a weighted eigenvalue problem of the beta-biased infinity Laplacian operator arising from the beta-biased tug-of-war. We characterize the principal eigenvalue by the comparison principle and show that beta-biased infinity Laplacian operator possesses two principal eigenvalues, corresponding to a positive and a negative principal eigenfunction. When a parameter is less than the principal eigenvalue, certain existence and uniqueness results of the inhomogeneous equations related to this problem are established. As an application, we obtain the decay estimates for viscosity solutions of the parabolic problem associated to the beta-biased infinity Laplacian. In the process, we also establish the Lipschitz regularity and Harnack inequality by barrier method.