(untitled) (Analysis in Theory and Applications, 2023)
In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the β−biased infinity Laplacian equation ∆^β_∞u = 0, where β ∈ R is a fixed constant and ∆^β_∞u := ∆^N_∞u + β|Du|, which arises from the random game named biased tug-of-war. By studying directly the β−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.