Elliptic equations in graphs via stochastic games

Consider a connected finite graph E with set of vertices X . Choose a nonempty subset Y ⊂ X , not equal to the whole X , and call it the boundary Y = ∂ X . We are given a real valued function F : Y → R . Our objective is to find function u on X , such that u = F on Y and u satisfies the following equation for all x ∈ X Y ux=amax y∈Sxu y+bmin y∈Sxu y+g y∈Sxu y#S x ,1 where alpha, beta, and gamma are some predetermined non-negative constants such that alpha + beta + gamma = 1, for x ∈ X , S(x) is the set of vertices connected to x by an edge, and ;Keywords: p-harmonic function, infinity harmonic function, p-harmonious function, stochastic games, unique continuation.