p-harmonious functions with drift on graphs via games

Abstract

In a connected finite graph $E$ with set of vertices $mathfrak{X}$, choose a nonempty subset, not equal to the whole set, $Ysubset mathfrak{X}$, and call it the boundary $Y=partialmathfrak{X}$. Given a real-valued function $F: Yo mathbb{R}$, our objective is to find a function $u$, such that $u=F$ on $Y$, and for all $xin mathfrak{X}setminus Y$, $$ u(x)=alpha max_{y in S(x)}u(y)+eta min_{y in S(x)}u(y) +gamma Big( frac{sum_{y in S(x)}u(y)}{#(S(x))}Big). $$ Here $alpha, eta, gamma $ are non-negative constants such that $alpha+eta + gamma =1$, the set $S(x)$ is the collection of vertices connected to $x$ by an edge, and $#(S(x))$ denotes its cardinality. We prove the existence and uniqueness of a solution of the above Dirichlet problem and study the qualitative properties of the solution.

Keywords

• Dirichlet problem
• comparison principle
• mean-value property
• stochastic games
• unique continuation