On the definition and properties of $p$-harmonious functions

Abstract

We consider functions that satisfy the identity u ε (x) = α 2 sup B ε (x) u ε + inf B ε (x) u ε + β B ε (x) u ε dy for a bounded domain in R n. Here ε > 0 and α, and β are suitable nonnega-tive coefficients such that α + β = 1. In particular, we show that these functions are uniquely determined by their boundary values, approximate p-harmonic functions for 2 ≤ p < ∞ (for a choice of p that depends on α and β), and satisfy the strong comparison principle. We also analyze their relation to the theory of tug-of-war games with noise. Mathematics Subject Classification (2010): 91A15 (primary); 35B50, 35J25, 35J70, 49N70, 91A24 (secondary).