\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage[all]{xy} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 114, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/114\hfil $p$-harmonious functions] {$p$-harmonious functions with drift on graphs via games} \author[A. P. Sviridov\hfil EJDE-2011/114\hfilneg] {Alexander P. Sviridov} % in alphabetical order \address{Alexander P. Sviridov \newline Department of Mathematics\\ University of Pittsburgh, Pittsburgh, PA 15260, USA} \email{aps14@pitt.edu} \thanks{Submitted October 26, 2010. Published September 6, 2011.} \subjclass[2000]{35Q91, 35B51, 34A12, 31C20} \keywords{Dirichlet problem; comparison principle; mean-value property; \hfill\break\indent stochastic games; unique continuation} \begin{abstract} In a connected finite graph $E$ with set of vertices $\mathfrak{X}$, choose a nonempty subset, not equal to the whole set, $Y\subset \mathfrak{X}$, and call it the boundary $Y=\partial\mathfrak{X}$. Given a real-valued function $F: Y\to \mathbb{R}$, our objective is to find a function $u$, such that $u=F$ on $Y$, and for all $x\in \mathfrak{X}\setminus Y$, $$u(x)=\alpha \max_{y \in S(x)}u(y)+\beta \min_{y \in S(x)}u(y) +\gamma \Big( \frac{\sum_{y \in S(x)}u(y)}{\#(S(x))}\Big).$$ Here $\alpha, \beta, \gamma$ are non-negative constants such that $\alpha+\beta + \gamma =1$, 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. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} %\newcommand{-\kern-1.1em\int\nolimits}{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -1.1em \int \nolimits} %\newcommand{-\kern-1.1em\int\nolimits}{-\kern-1.1em\int\nolimits} \section{Introduction} \label{intro} The goal of this paper is to study functions that satisfy $$\label{p-harm-drift} u(x)=\alpha \max_{y \in S(x)}u(y)+\beta \min_{y \in S(x)}u(y) +\gamma \Big( \frac{\sum_{y \in S(x)}u(y)}{\#(S(x))}\Big).$$ We denote a graph by $E$ and the collection of vertices by $\mathfrak{X}$. We choose $Y$ to be a proper nonempty subset of $\mathfrak{X}$ and call it the boundary. In equation \eqref{p-harm-drift} the set $S(x)$ is the collection of vertices connected to the given vertex $x$ by a single edge, and $\alpha$, $\beta$ and $\gamma$ are predetermined non-negative constants such that $\alpha + \beta + \gamma=1$. The cardinality of $S(x)$ is denoted by $\# S(x)$ . A function satisfying \eqref{p-harm-drift} is called $p$-harmonious with drift, by analogy with continuous case studied in \cite{MPR2}. Functions of this type arise as approximations of $p$-harmonic functions. In particular, an approximating sequence could be generated by running zero-sum stochastic games on a graph of decreasing step-size. The value of the game function satisfies a nonlinear equation, which is directly linked to the existence and uniqueness of the solution of the $p$-Laplacian as demonstrated in \cite{PSSW,PS,MPR1}. We present the connections between equation \eqref{p-harm-drift} and game theory in Theorem \ref{MartingalePDETheor}. We formally pose the Dirichlet problem: For a given $F:Y\to \mathbb{R}$ find a function $u$ defined on $\mathfrak{X}$, such that $u=F$ on $Y$ and $u$ satisfies \eqref{p-harm-drift}. We address questions of existence and uniqueness of the solution of this Dirichlet problem in Theorems \ref{existence} and \ref{cmpr}. We state the strong comparison principle in Theorem \ref{strcomppricip}. We also study the question of unique continuation for $p$-harmonious functions with drift. In particular we present an example of $p$-harmonious function which does not have the unique continuation property. The current manuscript is based on the results obtained in \cite{APS}. The equation \eqref{p-harm-drift} can be restated in a more traditional notation with the help of the following definitions, which we borrowed from \cite{KLW}. \begin{definition} \label{def1.1} \rm The Laplace operator on the graph is given by $$\Delta u(x)=-\kern-1.1em\int\nolimits_{S(x)}u-u(x).$$ \end{definition} \begin{definition} \label{def1.2} \rm The infinity Laplacian on the graph is given by $$\Delta_{\infty}u(x)=\frac{1}{2}(\max_{S(x)} u+\min_{S(x)}u)-u(x).$$ \end{definition} \begin{definition} \label{def1.3} \rm For $X=(x,y,z)\in \mathbb{R}^3$ we define the analog of the maximal directional derivative $$\langle X\rangle_{\infty}=\max \{x,y,z\}.$$ \end{definition} With the above definitions we can restate \eqref{p-harm-drift} as $$(\alpha-\beta)\langle \nabla u \rangle_{\infty} +2\beta \Delta_{\infty}u+\gamma \Delta u=0.$$ \section{Game setup and definitions} \label{sec:2} Most of our results are proved using the following game. We consider a connected graph $E$ with vertex set $\mathfrak{X}$. The set $\mathfrak{X}$ is finite unless stated otherwise. We equip $\mathfrak{X}$ with the $\sigma$-algebra $\mathcal{F}$ of all subsets of $\mathfrak{X}$. For an arbitrary vertex $x$ we define $S(x)$ the collection of vertices, which are connected to the vertex $x$ by a single edge. In case $\mathfrak{X}$ is infinite, we require that $\mathfrak{X}$ is at least locally finite; i.e. the cardinality of $S(x)$ is finite. At the beginning of the game a token is placed at some point $x_0 \in \mathfrak{X}$. Then we toss a three-sided virtual coin. The side of a coin labelled 1 comes out with probability $\alpha$ and in this case player I chooses where to move the token among all vertices in $S(x)$. The side of a coin labelled 2 comes out with probability $\beta$ and in this case player II chooses where to move the token among all vertices in $S(x)$. Finally, the side of a coin labelled 3 comes out with probability $\gamma$ and in this case we choose the next point randomly (uniformly) among all vertices in $S(x)$. This setup has been described in \cite{PSSW} and in \cite{PPS} and is known as \lq\lq biased tug-of-war with noise\rq\rq. The game stops once we hit the boundary set $Y$. The set $Y$ is simply predetermined non-empty set of vertices at which game terminates. In the game literature the set $Y$ is called set of absorbing states. Let $F:Y\to \mathbb{R}$ be the payoff function defined on $Y$. If game ends at some vertex $y \in Y$, then player I receives from player II the sum of $F(y)$ dollars. Let us define the value of the game for player I. Firstly, we formalize the notion of a pure strategy. We define a strategy $S_I$ for player I as a collection of maps $\{\sigma^{k}_I\}_{k\in \mathbb{N}}$, such that for each $k$, \begin{gather*} \sigma^{k}_I: \mathfrak{X}^k \to \mathfrak{X},\\ \sigma^{k}_I(x_0, \dots , x_{k-1})= x_k, \end{gather*} where $$\mathfrak{X}^k=\underbrace{\mathfrak{X}\times\mathfrak{X} \times\dots \times\mathfrak{X}}_{k\, \text{times}}.$$ Hence, $\sigma^{k}_I$ tells player I where to move given $(x_0, \dots , x_{k-1})$ - the history of the game up to the step $k$, if he wins the toss. We call a strategy {\it stationary} if it depends only on the current position of the token. Given two strategies for player I and II the transition probabilities for $k\geq 1$ are given by $\pi_{k}(x_0,\dots ,x_{k-1};y)= \alpha \delta_{\sigma_I^{k}(x_0,\dots ,x_{k-1})}(y)+\beta \delta_{\sigma_{II}^{k}(x_0,\dots ,x_{k-1})}(y) +\gamma U_{S(x_{k-1})}(y),$ where we have set $$U_{S(x_{k-1})} \text{ is a uniform distribution on } S(x_{k-1}) \text{ and } \pi_{0}(y)=\delta_{x_0}(y).$$ We equip $\mathfrak{X}^k$ with product $\sigma$-algebra $\mathcal{F}^k$, $$\mathcal{F}^k=\underbrace{\mathcal{F}\otimes\mathcal{F}\otimes\dots \otimes\mathcal{F}}_{k\, \text{times}}$$ and then we define a probability measure on $(\mathfrak{X}^k, \mathcal{F}^k)$ in the following way: \begin{gather*} \mu_0= \pi_0=\delta_{x_0}, \\ \mu_k(A^{k}\times A)=\int_{A^{k}}\pi_{k}(x_0,\dots ,x_{k-1};A) d\mu_{k-1}, \end{gather*} where $A^{k-1}\times A$ is a rectangle in $(\mathfrak{X}^k, \mathcal{F}^{k})$. The space of infinite sequences with elements from $\mathfrak{X}$ is $\mathfrak{X}^{\infty}$. Let $X_k:\mathfrak{X}^{\infty}\to \mathfrak{X}$ be the coordinate process defined by $$X_k(h)=x_k, \quad \text{for } h=(x_0, x_1, x_2, x_3,\dots ) \in \mathfrak{X}^{\infty}.$$ We equip $\mathfrak{X}^{\infty}$ with product $\sigma$-algebra $\mathcal{F}^{\infty}$. For precise definition of $\mathcal{F}^{\infty}$ see \cite{KALLENBERG}. The family of $\{\mu_k\}_{k\geq 0}$ satisfies the conditions of Kolmogorov extension theorem \cite{VARADHAN}, therefore, we can conclude that there exists a unique measure $\mathbb{P}^{x_0}$ on $(\mathfrak{X}^{\infty}, \mathcal{F}^{\infty})$ with the following property: $$\mathbb{P}^{x_0}(B_k\times \mathfrak{X}\times \mathfrak{X}\times \times\mathfrak{X} \dots )=\mu_k(B_k), \quad \text{for } B_k \in \mathcal{F}^k$$ and $$\mathbb{P}^{x_0}[X_k\in A|X_0=x_0, X_1=x_1,\dots , X_{k-1} =x_{k-1}]=\pi_{k}(x_0,\dots ,x_{k-1};A).$$ We are now ready to define the value of the game for player I. The boundary hitting time is given by $$\tau=\inf_{k}\{X_k \in Y\}.$$ Consider strategies $S_I$ and $S_{II}$ for player I and player II respectively. We define $$F^x_{-}(S_I, S_{II}) =\begin{cases} \mathbb{E}^x_{S_I,S_{II}}[F(X_{\tau})] & \text{if } \mathbb{P}^x_{S_I,S_{II}}(\tau<\infty)=1\\ -\infty &\text{otherwise} \end{cases}$$ $$F^x_{+}(S_I, S_{II}) =\begin{cases} \mathbb{E}^x_{S_I,S_{II}}[F(X_{\tau})] & \text{if } \mathbb{P}^x_{S_I,S_{II}}(\tau<\infty)=1\\ +\infty &\text{otherwise} \end{cases}$$ The value of the game for player I is $$u_I(x)=\sup_{S_I}\inf_{S_{II}}{\mathbb F}^x_{-}(S_I,S_{II})$$ and the value of the game for player II is $$u_{II}(x)=\inf_{S_{II}}\sup_{S_I}{\mathbb F}^x_{+}(S_I,S_{II})$$ These definitions penalize players severely for not being able to force the game to end. Whenever player I has a strategy to finish the game almost surely, then we simplify notation by setting $$u_I(x)=\sup_{S_I}\inf_{S_{II}}E^x_{S_I,S_{II}}[F(X_{\tau})].$$ Similarly, for player II we set $$u_{II}(x)=\inf_{S_{II}}\sup_{S_I}E^x_{S_I,S_{II}}[F(X_{\tau})].$$ The following lemma states rigorously whether player I has a strategy to finish the game almost surely: \begin{lemma} \label{lem2.1} If $\mathfrak{X}$ is a finite set, then player I (player II) has strategies to finish the game almost surely. \end{lemma} \begin{proof} When $\gamma=0$, this result was already proven by Peres, Schramm, Sheffield, and Wilson in \cite[Theorem 2.2]{PSSW}. When $\gamma \neq 0$, the statement follows from the fact that random walk on a finite graph is recurrent. \end{proof} We always have $u_I(x)\leq u_{II}(x)$. Whenever $u_I(x)= u_{II}(x)$ for all $x \in \mathfrak{X}$ we say that game has a value. \section{Existence} \label{sec:3} Here is the first existence result for equation \eqref{p-harm-drift}. \begin{theorem} [Dynamic Programming Principle equals Mean Value Property] \label{existence} The value functions $u_I$ and $u_{II}$ satisfy the Dynamic Programming Principle (DPP) or the Mean Value Property (MVP): \begin{gather}\label{mvpdpp} u_I(x)=\alpha \max_{y \in S(x)}u_I(y) +\beta \min_{y \in S(x)}u_I(y) +\gamma -\kern-1.1em\int\nolimits_{S(x)}u_I(y)dy, \\ u_{II}(x)=\alpha \max_{y \in S(x)}u_{II}(y) +\beta \min_{y \in S(x)}u_{II}(y) +\gamma -\kern-1.1em\int\nolimits_{S(x)}u_{II}(y)dy. \end{gather} \end{theorem} The above result is true in the general setting of discrete stochastic games (see Maitra and Sudderth, \cite[chapter 7]{MS}). Here we provide a simpler proof in Markovian case. It turns out that optimal strategies are Markovian (see \cite[chapter 5]{MS}). % \begin{proposition}[The stationary case]\label{DPPMarkovian} In a game with stationary strategies the value functions $u_I$ and $u_{II}$ satisfy the Dynamic Programming Principle (DPP) or the Mean Value Property (MVP): \begin{gather} u_I(x)=\alpha \max_{y \in S(x)}u_I(y)+\beta \min_{y \in S(x)}u_I(y) +\gamma -\kern-1.1em\int\nolimits_{S(x)}u_I(y)dy, \\ u_{II}(x)=\alpha \max_{y \in S(x)}u_{II}(y) +\beta \min_{y \in S(x)}u_{II}(y) +\gamma -\kern-1.1em\int\nolimits_{S(x)}u_{II}(y)dy. \end{gather} \end{proposition} \begin{proof} We will provide a proof only for $u_I$; the proof for $u_{II}$ follows by symmetry. Take a set of vertices $\mathfrak{X}$, boundary $Y$ and adjoin one vertex $y^*$ to the boundary. Denote new boundary by $Y^*=Y\cup \{y^*\}$ and the new set of vertices by $\mathfrak{X}^*= \mathfrak{X}\setminus\{y^*\}$ and define $$F^*(y) =\begin{cases} F(y) & \text{if } y \in Y\\ u_I(y^*)&\text{if } y= y^*. \end{cases}$$ Let $u_I(x)$ be the value of the game with $\mathfrak{X}$ and $Y$, and $u^{*}_I(x)$ be the value of the game with $\mathfrak{X}^*$ and $Y^*$. The goal is to show that $$u^{*}_I(x)=u_I(x).$$ Once we prove the above, the main result follows by extending $F$ to the set $S(x)$. \begin{remark} \label{rmk3.3} \rm The idea of extending $F$ is used in \cite[Lemma 3.5]{PSSW} \end{remark} Hence, we have to show $u^{*}_I(x)=u_I(x)$. Since we consider only Markovian strategies we can think of them as mappings $S_I:\mathfrak{X}\to \mathfrak{X}$. For the game $\mathfrak{X}^*$ and $Y^*$, we define $S^{*}_I$ as a restriction of $S_I$ to $\mathfrak{X}^*$ Here are the steps in detail: $$\begin{split} u^{*}_I(x) &= \sup_{S_I^{*}}\inf_{S_{II}^{*}}\big (E^x_{S_I^{*}, S_{II}^{*}}F^*(X_{\tau^*})\big )\\ &=\sup_{S_I^{*}}\inf_{S_{II}^{*}}\big(E^x_{S_I^{*}, S_{II}^{*}}F^*(X_{\tau^*})\chi_{\{X_{\tau^*}=y^*\}} +E^x_{S_I^{*},S_{II}^{*}}F^*(X_{\tau^*})\chi_{\{X_{\tau^*}=y^*\}^c} \big )\\ &=\sup_{S_I^{*}}\inf_{S_{II}^{*}}\big (E^x_{S_I^{*}, S_{II}^{*}}u_I(y^*)\chi_{\{X_{\tau^*}=y^*\}}+E^x_{S_I^{*}, S_{II}^{*}}F^*(X_{\tau^*})\chi_{\{X_{\tau^*}=y^*\}^c}\big )\\ &= \sup_{S_I^{*}}\inf_{S_{II}^{*}}\big (E^x_{S_I^{*}, S_{II}^{*}}\sup_{S_I}\inf_{S_{II}}E^{y^*}_{S_I, S_{II}}F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}}\\ &\quad +E^x_{S_I^{*},S_{II}^{*}}F^*(X_{\tau^*}) \chi_{\{X_{\tau^*}=y^*\}^c}\big )\\ &= \sup_{S_I^{*}}\inf_{S_{II}^{*}} \sup_{S_I}\inf_{S_{II}} \big (E^x_{S_I^{*},S_{II}^{*}}\big (E^{y^*}_{S_I, S_{II}}F(X_{\tau})\big )\chi_{\{X_{\tau^*}=y^*\}}\\ &\quad +E^x_{S_I^{*},S_{II}^{*}}F^*(X_{\tau^*}) \chi_{\{X_{\tau^*}=y^*\}^c}\big ). \end{split}$$ If we can show that $$\label{stationaryproof} \begin{split} &\sup_{S_I^{*}}\inf_{S_{II}^{*}} \sup_{S_I}\inf_{S_{II}} \big (E^x_{S_I^{*},S_{II}^{*}}\big (E^{y^*}_{S_I, S_{II}} F(X_{\tau})\big )\chi_{\{X_{\tau^*}=y^*\}}\\ &+E^x_{S_I^{*},S_{II}^{*}}F^*(X_{\tau^*})\chi_{\{X_{\tau^*}=y^*\}^c} \big )\\ &= \sup_{S_I^{*}}\inf_{S_{II}^{*}} \sup_{S_I}\inf_{S_{II}} \big (E^x_{S_I,S_{II}} F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}} +E^x_{S_I,S_{II}}F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}^c}\big ). \end{split}$$ We can complete the proof in the following way: \begin{align*} u^{*}_I(x) &= \sup_{S_I^{*}}\inf_{S_{II}^{*}} \sup_{S_I}\inf_{S_{II}} \big (E^x_{S_I,S_{II}} F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}} +E^x_{S_I,S_{II}}F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}^c}\big )\\ & = \sup_{S_I}\inf_{S_{II}}\sup_{S_I^{*}}\inf_{S_{II}^{*}} \big (E^x_{S_I,S_{II}} F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}} +E^x_{S_I,S_{II}}F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}^c}\big )\\ &= \sup_{S_I}\inf_{S_{II}}\big (E^x_{S_I,S_{II}} F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}}+E^x_{S_I,S_{II}} F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}^c}\big )\\ &= \sup_{S_I}\inf_{S_{II}}E^x_{S_I,S_{II}} F(X_{\tau})=u_I(x). \end{align*} Let us clarify \eqref{stationaryproof}. Actually, we have the following two equalities \begin{gather}\label{mar11} E^x_{S_I^{*},S_{II}^{*}}E^{y^*}_{S_I, S_{II}} F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}}=E^x_{S_I,S_{II}} F(X_{\tau})\chi_{\{X_{\tau^*}=y^*\}}, \\ \label{mar2}E^x_{S_I^{*},S_{II}^{*}}F^*(X_{\tau^*}) \chi_{\{X_{\tau^*}=y^*\}^c}=E^x_{S_I,S_{II}}F(X_{\tau}) \chi_{\{X_{\tau^*}=y^*\}^c} \end{gather} Equation ($\ref{mar11}$) could be thought of as payoff computed for the trajectories that travel through a point $y^*$. Roughly speaking we first discount boundary points to the point $y^*$ and then discount value at $y^*$ back to $x$ which is the same as to discount boundary points to $x$ through trajectories that contain $y^*$, keeping in mind that $S^{*}_{i}$ is just a restriction of $S_{i}$. Equation \eqref{mar2} is a payoff computed for the trajectories that avoid $y^*$, and, therefore, there is no difference between $S^{*}_{i}$ and $S_{i}$, since $S^{*}_{i}$ is just a restriction of $S_{i}$ to $\mathfrak{X}\setminus \{y^*\}$. \end{proof} The following proposition is an extension of the result stated in \cite{MPR3}. It characterizes optimal strategies. By optimal strategies we mean any pair of strategies $\hat{S}_I$ and $\hat{S}_{II}$ such that $$E^x_{\hat{S}_I,\hat{S}_{II}}F(X_{\tau}) =\sup_{S_I}\inf_{S_{II}}E^x_{S_I,S_{II}}F(X_{\tau}) =u_I=u_{II}.$$ \begin{proposition}\label{dppsimple} Consider a game on the graph $E$ with finite set of vertices $\mathfrak{X}$. Then the the strategy $\hat{S}_I$ ($\hat{S}_{II})$ under which player I (player II) moves from vertex $x$ to vertex $z$ with $$u(z)=\max_{y\in S(x)} u(y), \quad (u(z)=\min_{y\in S(x)} u(y))$$ is optimal. \end{proposition} \begin{proof} Let us start the game at vertex $x$ ($X_0=x$). We claim that under strategies $\hat{S}_I$ and $\hat{S}_{II}$ $u_I(X_k)$ is a martingale due to following arguments: $$\begin{split} &\mathbb{E}^x_{ \hat{S}_I, \hat{S}_{II}}[u_I(X_k)|X_0,\dots ,X_{k-1}]\\ &=\alpha u_I(X^I_{k})+\beta u_I(X^{II}_{k}) +\gamma -\kern-1.1em\int\nolimits_{S(X_{k-1})}u_I(y)dy \\ & = \alpha \max_{y \in S(X_{k-1})}u_I(y) +\beta \min_{y \in S(X_{k-1})}u_I(y) +\gamma -\kern-1.1em\int\nolimits_{S(X_{k-1})}u_I(y)dy\\ &= u_I(X_{k-1}), \end{split}$$ where $v(X^{I}_{k})$ indicates the choice of player I and $v(X^{II}_{k})$ indicates the choice of player II. Then $u_I(X^{II}_{k})=\min_{y \in S(X_{k-1})}u_I(y),\quad u_I(X^{II}_{k})=\max_{y \in S(X_{k-1})}u_I(y)$ by choice of strategies $\hat{S}_I$ and $\hat{S}_{II}$. In addition, since $u_I$ is a bounded function, we conclude that $u_I(X_k)$ is a uniformly integrable martingale. Hence, by Doob's Optional Stopping Theorem $$E^x_{\hat{S}_I,\hat{S}_{II}}F(X_{\tau}) =E^x_{\hat{S}_I,\hat{S}_{II}}u_I(X_{\tau}) =E^x_{\hat{S}_I,\hat{S}_{II}}u_I(X_{0}) =u_I(x),$$ \end{proof} \begin{example} \label{examp3.5} \rm We would like to warn the reader that the Proposition \ref{dppsimple} does not claim that tugging towards that maximum of $F$ on the boundary would be an optimal strategy for player I. Figure \ref{fig1} shows a counterexample. \begin{figure}[ht] $$\xymatrix{ -1 \\ e_0 \ar@{<->}[u] &e_1 \ar@{<->}[l] &3/2\ar@{<->}[l] \\ 1 \ar@{<->}[u] }$$ \caption{Counterexample - tugging towards the boundary} \label{fig1} \end{figure} The boundary vertices are indicated by the numbers, which reflect the value of $F$ at each vertex. We consider the game starting at vertex $e_0$ and require player II always pull towards the vertex labelled -1. For player I we choose $S^{a}_I$ to be the strategy of always tugging towards vertex $3/2$ and let $S^{b}_I$ be the strategy of moving towards vertex 1. We see that \begin{gather} E^{e_0}_{S^{a}_I, S_{II}}F(X_{\tau})=-1\cdot 2/3+3/2\cdot 1/3 =-1/6, \\ E^{e_0}_{S^{b}_I, S_{II}}F(X_{\tau})=-1\cdot 1/2+1\cdot 1/2=0\,. \end{gather} \end{example} \section{Uniqueness}\label{sec:4} Uniqueness will follow from the comparison principle below proven by using Doob's Optional Sampling Theorem. \begin{theorem}[via Martingales]\label{cmpr} Let $v$ be a solution of $$\label{uniqueness1} v(x)=\alpha \max_{y \in S(x)}v(y)+\beta \min_{y \in S(x)}v(y) +\gamma -\kern-1.1em\int\nolimits_{S(x)}v(y)dy$$ on a graph $E$ with a countable set of vertices $\mathfrak{X}$ and boundary $Y$. Assume \begin{itemize} \item $F(y)=u_I(y)$, \text{ for all } $y \in Y$, \item $\inf_{Y}F>-\infty$, \item $v$ bounded from below, and \item $v(y)\geq F(y), \quad \text{ for all } y \in Y$ \end{itemize} Then $u_I$ is bounded from below on $\mathfrak{X}$ and $v(x)\geq u_I(x)$, for $x \in \mathfrak{X}$. \end{theorem} \begin {proof} Note that we only need $\leq$'' in equation \eqref{uniqueness1}. The theorem says that $u_I$ is the smallest super-solution with given boundary value $F$. We proceed as in \cite[Lemma 2.1]{PSSW}. Since the game ends almost surely, $$u_I\geq \inf_{Y}F >-\infty$$ which proves that $u_I$ is bounded from below. Now we have to show that $$v(x)\geq \sup_{S_I}\inf_{S_{II}}F_{-}^x(S_I, S_{II})=u_I(x)$$ If we fix an arbitrary strategy $S_I$, then we have to show that $$\label{compineq} v(x)\geq \inf_{S_{II}}F_{-}^x(S_I, S_{II}).$$ Consider a game that start at vertex $x$ ($X_0=x$). We have two cases \textbf{Case 1:} If our fixed $S_I$ cannot force the game to end a.s. (i.e. $\mathbb{P}^x_{S_I, S_{II}}(\tau < \infty)<1$), then by the definition of $F_{-}$, $\inf_{S_{II}}F_{-}^x(S_I, S_{II})=-\infty$ and the inequality \eqref{compineq} holds. \textbf{Case 2:} Now assume that our fixed $S_I$ forces the game to end despite all the efforts of the second player. Let player II choose a strategy of moving to $\min_{y\in S(x)}v(y)$ - denote such a strategy $\hat{S}_{II}$. If we prove that $v(X_k)$ is a supermartigale bounded from below, then we can finish the proof by applying Doob's Optional Stopping Theorem: \begin{align*} \inf_{S_{II}}\mathbb{E}^x_{S_I, S_{II}}F(X_\tau) & \leq \mathbb{E}^x_{S_I, \hat{S}_{II}}F(X_\tau) \leq \mathbb{E}^x_{S_I, \hat{S}_{II}}v(X_\tau)\\ & \leq \mathbb{E}^x_{S_I, \hat{S}_{II}}v(X_0) = v(X_0)=v(x), \end{align*} where we have used Fatou's lemma. The result follows upon taking $\sup_{S_I}$. Hence, we only need to prove that $v(X_k)$ is a supermartingale under measure $\mathbb{P}^x_{S_I, \hat{S}_{II}}$: \begin{align*} &\mathbb{E}^x_{S_I, \hat{S}_{II}}[v(X_k)|X_0,\dots ,X_{k-1}]\\ &= \alpha v(X^I_{k})+\beta v(X^{II}_{k}) +\gamma -\kern-1.1em\int\nolimits_{S(X_{k-1})}v(y)dy \\ &\leq \alpha \max_{y \in S(X_{k-1})}v(y) +\beta \min_{y \in S(X_{k-1})}v(y) +\gamma -\kern-1.1em\int\nolimits_{S(X_{k-1})}v(y)dy = v(X_{k-1}), \end{align*} where $v(X^{I}_{k})$ indicates the choice of player I and $v(X^{II}_{k})$ indicates the choice of player II. Then $v(X^{II}_{k})=\min_{y \in S(X_{k-1})}v(y)$ by choice of strategy for player II. \end{proof} In case $\min_{y \in S(X_{k-1})}v(y)$ is not achieved (i.e. graph is not locally finite), we need to modify the above proof by making player II move within $\epsilon$ neighborhood of $\min_{y \in S(X_{k-1})}v(y)$. We can prove similar result for $u_{II}$. The next theorem is the extension of the result obtained in \cite{MPR2}. \begin{theorem}\label{valueexists} If graph $E$ is finite and $F$ is bounded below on $Y$, then $u_I=u_{II}$, so the game has a value. \end{theorem} \begin{proof} Clearly, finite $E$ implies that $F$ is bounded below. We included this redundant statement to suggest future possible extensions to an uncountable graph. We know that $u_I\leq u_{II}$ always holds, so we only need to show $u_I\geq u_{II}$. Assume F is bounded below. Similar to the proof of Lemma \ref{cmpr} we can show that $u_I$ is a supermartingale bounded below by letting player I to choose an arbitrary strategy $S_I$ and requiring player II always move to $\min_{y\in S(x)}u_I(y)$ from $x$ - strategy $\hat{S}_{II}$. For simplicity of the presentation we consider a case when $\min_{y\in S(x)}u_I(y)$ is achievable, for the general case we have to employ $\epsilon$, like in Theorem $\ref{cmpr}$. We start the game at $x$, so $X_0=x$. Recall $u_{II}(x)=\inf_{S_{II}}\sup_{S_I}F_{+}(S_I, S_{II})$ \begin{align*} u_{II}(x) & \leq \sup_{S_I}\mathbb{E}^x_{S_I,\hat{S}_{II}}[F(X_{\tau})] \quad \text{(since E is finite)}\\ &=\sup_{S_I}\mathbb{E}^x_{S_I, \hat{S}_{II}}[u_I(X_{\tau})]\\ & \leq \sup_{S_I}\mathbb{E}^x_{S_I, \hat{S}_{II}}[u_I(X_{0})] =u_I(x). \end{align*} Due to Doob's Optional Stopping Theorem. \end{proof} \section{Connections among games, partial differential equations and DPP} \label{sec:5} This section summarizes some previous results and presents new prospectives on known issues. \begin{theorem}\label{MartingalePDETheor} Assume we are given a function $u$ on the set of vertices $\mathfrak{X}$ and consider a strategy $\hat{S}_I$ ($\hat{S}_{II})$ where player I (player II) moves from vertex $x$ to vertex $z$ with $$u(z)=\max_{y\in S(x)} u(y) \quad (u(z)=\min_{y\in S(x)} u(y)).$$ Then the following two statements are equivalent: \begin{itemize} \item the process $u(X_n)$ is a martingale under the measure induced by strategies $\hat{S}_I$ and $\hat{S}_{II}$, \item the function $u$ is a solution of Dirichlet problem \eqref{p-harm-drift}. \end{itemize} In addition, $u(X_n)$ is a martingale under the measure induced by strategies $\hat{S}_I$ and $\hat{S}_{II}$ implies that $\hat{S}_I$ and $\hat{S}_{II}$ are the optimal strategies. \end{theorem} \begin{proof} Suppose that $u(X_n)$ is a martingale under measure induced by strategies $\hat{S}_I$ and $\hat{S}_{II}$. Fix an arbitrary point $x \in \mathfrak{X}$ and consider a game which starts at $x=X_0$, then $$\label{MartingalePDE} \begin{split} E^x_{\hat{S}_I, \hat{S}_{II}}[u(X_1)|X_0] &=\alpha u(X^{I}_1)+\beta u(X^{II}_1)+\gamma-\kern-1.1em\int\nolimits_{S(X_0)}u(y)dy \\ &=\alpha \max_{y\in S(X_0)} u(y)+\beta \min_{y\in S(X_0)} +\gamma-\kern-1.1em\int\nolimits_{S(X_0)}u(y)dy \\ &=u(X_0). \end{split}$$ Conversely, assume that $u$ solves Dirichlet problem \eqref{p-harm-drift}, then \eqref{MartingalePDE} implies that $u(X_n)$ is a martingale under measure induced by strategies $\hat{S}_I$ and $\hat{S}_{II}$. Let us show a final implication. The result relies on the fact that our game has a value and value of game function is the solution of the Dirichlet problem \eqref{p-harm-drift}. Since $u(X_n)$ is a martingale under measure induced by strategies $\hat{S}_I$ and $\hat{S}_{II}$ we have $$E^x_{\hat{S}_I,\hat{S}_{II}}F(X_{\tau}) =E^x_{\hat{S}_I,\hat{S}_{II}}u(X_{\tau}) =E^x_{\hat{S}_I,\hat{S}_{II}}u(X_0)=u(x).$$ By the uniqueness result (Theorem \ref{cmpr}) $$u(x)=\sup_{S_I}\inf_{S_{II}}E^x_{S_I,S_{II}}F(X_{\tau}).$$ \end{proof} \section{Strong comparison principle}\label{sec:6} \begin{theorem}\label{strcomppricip} Assume that $u$ and $v$ are solutions of equation \eqref{p-harm-drift} on $\mathfrak{X}\setminus Y$, $\gamma \neq 0$, $u\leq v$ on the boudary $Y$, and exists $x\in \mathfrak{X}$ such that $u(x)=v(x)$, then $u=v$ through the whole $\mathfrak{X}$. \end{theorem} \begin{proof} By Theorem \ref{cmpr} from the fact that $u\leq v$ on the boundary we know that $u\leq v$ on $\mathfrak{X}$. By definition of $p$-harmonious function we have \begin{gather} v(x)=\alpha \max_{y \in S(x)}v(y)+\beta \min_{y \in S(x)}v(y) +\gamma -\kern-1.1em\int\nolimits_{S(x)}v(y)dy, \\ u(x)=\alpha \max_{y \in S(x)}u(y)+\beta \min_{y \in S(x)}u(y) +\gamma -\kern-1.1em\int\nolimits_{S(x)}u(y)dy. \end{gather} Since $u\geq v$ on $\mathfrak{X}$ we know that \begin{gather*} \max_{y \in S(x)}v(y)\leq \max_{y \in S(x)}u(y),\\ \min_{y \in S(x)}v(y)\leq \min_{y \in S(x)}u(y),\\ -\kern-1.1em\int\nolimits_{S(x)}v(y)dy\leq -\kern-1.1em\int\nolimits_{S(x)}u(y)U(dy). \end{gather*} But since $u(x)=v(x)$, we actually have equalities \begin{gather*} \max_{y \in S(x)}v(y)= \max_{y \in S(x)}u(y),\\ \min_{y \in S(x)}v(y)= \min_{y \in S(x)}u(y),\quad -\kern-1.1em\int\nolimits_{S(x)}v(y)dy= -\kern-1.1em\int\nolimits_{S(x)}u(y)dy. \end{gather*} From equality of average values and the fact that $u\geq v$ we conclude that $u=v$ on $S(x)$. Since our graph is connected, we immediately get the result. \end{proof} \section{Remarks on unique continuation} \label{uniquecont} We can pose the following question. Let $E$ be a finite graph with the vertex set $\mathfrak{X}$ and let $B_R(x)$ be the ball of radius $R$ contained within this graph. Here we assign to every edge of the graph length one and let $$d(x,y)=\inf_{x\sim y}\{|x\sim y|\},$$ where $x \sim y$ is the path connecting vertex $x$ to the vertex $y$ and $|x\sim y|$ is the number of edges in this path. Assume that $u$ is a $p$-harmonious function on $\mathfrak{X}$ and $u=0$ on $B_R(x)$. Does this mean that $u=0$ on $\mathfrak{X}$? It seems like the answer to this question depends on the values of $u$ on the boundary $Y$, as well as properties of the graph $E$ itself. Here we can provide simple examples for particular graph, which shows that $u$ does not have to be zero through the whole $\mathfrak{X}$. See tables \ref{table1} and \ref{table2}. \begin{table}[ht!] \caption{$p=2,\; 8$ neighbors} \label{table1} \begin{center} \begin{tabular}{|ccccccccccc|} \hline\\ 164 & $-349$ & 80 & 163 & 1& $-164$ &1 &163 & 96 & $-617$ &74 \\ $-349$ & $-52$ & $-19$ & 28 & 1& $-20$ & 1 & 28 & $-38$ & $-9$ & 596 \\ 80 & $-19$ & $-4$ & 1 & 1 & $-2$ & 1 & 1 & $-1$ & 35 & $-217$ \\ 163 & 28 & 1 & 0 & 0& 0 & 0 & 0 & 1 & $-26$ & $-26$ \\ 1 & 1 & 1 & 0 & 0& 0 & 0 & 0 & $-2$ & 1 & 1 \\ $-164$ & $-20$ & $-2$ & 0 & 0& 0 & 0 & 0 & 1 & 7 & 52\\ 1 & 1 & 1 & 0 & 0& 0 & 0 & 0 & 1 & 1 & 1\\ 163 & 28 & 1 & 0 & 0& 0 & 0 & 0 & $-2$ & 1 & $-53$\\ 80 & $-19$ & $-4$ & 1 & 1& $-2$ & 1 & 1 & $-1$ & $-19$ & 80\\ $-349$ & $-52$ & $-19$ & 28 & 1& $-20$ & 1 & 28 & $-19$ & 2 & $-160$\\ 164 & $-349$ & 80 & 163 & 1 & $-164$ & 1 & 163 & 77 & 403 & 461 \\ \hline \end{tabular} \end{center} \end{table} \begin{table}[ht!] \caption{$p=\infty,\, 8$ neighbors} \label{table2} \begin{center} \begin{tabular}{|ccccccccccc|} \hline\\ $-31$ & 21 & $-11$ & $-5$ & 1& 3 &1 &$-5$ & 11 & $-21$ & 23 \\ 21 & $-5$ & 5 & $-3$ & $-1$& 1 &$-1$ & 3 & $-5$ & 1 & 21 \\ $-11$ & 5 & 0 & 1 & $-1$ & 0 & 1 & $-1$ & 0 & 5 & $-11$ \\ $-5$ & $-3$ & 1 & 0 & 0& 0 & 0 & 0 & 1 & $-3$ & 5 \\ 3 & $-1$ & $-1$ & 0 & 0& 0 & 0 & 0 & $-1$ & $-1$ & 3 \\ 1 & 1 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 1 & 1\\ 3 & $-1$ & 1 & 0 & 0& 0 & 0 & 0 & 1 & $-1$ & 3\\ $-5$ & 3 & $-1$ & 0 & 0& 0 & 0 & 0 & $-1$ & 3 & $-5$\\ 11 & $-5$ & 0 & 1 & $-1$& 0 & 1 & $-1$ & 0 &$-5$ & 11\\ $-21$ & 1 & 5 & $-3$ & $-1$& 1 & $-1$ & 3 & $-5$ & 5 & $-21$\\ 23 & 21 & $-11$ & $-5$ & 1 & 3 & 1 & $-5$ & 11 & $-21$ & 31\\ \hline \end{tabular} \end{center} \end{table} \subsection*{Acknowledgements} The author would like to thank Dr. Manfredi for this his guidance, inspiration and support that made this research possible. \begin{thebibliography}{00} \bibitem{KLW} R. Kaufman, J. G. Llorente, J.-M. Wu; \emph{Nonlinear harmonic measures on trees}, Ann. Acad. Sci. Fenn. Math., 28, 279Ð302, (2003). \bibitem{KALLENBERG} O. Kallenberg; \emph{Foundations of Modern Probability}, 2-5. Springer, New York (1997) \bibitem{MS} A. P. Maitra, W. D. Sudderth; \emph{Discrete Gambling and Stochastic Games}, 171-226. Springer-Verlag, New York (1996). \bibitem{MPR1} J. J. Manfredi, M. Parviainen, J. D. Rossi; \emph{An asymptotic mean value characterization for $p$-harmonic functions}, Proc. Amer. Math. Soc., 138(3), 881-889 (2010). \bibitem{MPR2} J. J. Manfredi, M. Parviainen, J. D. Rossi; \emph{On the definition and properties of $p$-harmonious functions}, preprint (2009). \bibitem{MPR3} J. J. Manfredi, M. Parviainen, J. D. Rossi; \emph{Dynamic programming principle for tug-of-war games with noise}, preprint (2009). \bibitem{PPS} Yuval Peres, Gabor Pete, and Stephanie Somersille; \emph{Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones}, preprint (2009). \bibitem{PS} Y. Peres, S. Sheffield; \emph{Tug-of-war with noise: A game-theoretic view of the $p$-Laplacian}, Duke Math. J., 145 (1), 91-120 (2008). \bibitem{PSSW} Y. Peres, O. Schramm, S. Sheffield, D. Wilson; \emph{Tug-of-war and the infinity Laplacian}, J. Amer. Math. Soc., 22, 167-210 (2009). \bibitem{APS} Alexander P. Sviridov; \emph{Elliptic equations in graphs via stochastic games}, Dissertation (2010). \bibitem{VARADHAN} S. R. S. Varadhan; \emph{Probability Theory}, 41-42. Courant Institute of Mathematical Science, New York (2001). \end{thebibliography} \end{document}