\chapter{Probabilistic interpretations}
\label{sec:Probabilistic-interpretation}

\headerquote{From its shady beginnings devising gambling strategies and counting corpses in medieval London, probability theory and statistical inference now emerge as better foundations for scientific models, especially those of the process of thinking and as essential ingredients of theoretical mathematics, even the foundations of mathematics itself.}{---~David~Mumford}

\headerquote{God not only plays dice. He also sometimes throws the dice where they cannot be seen.}{---~S.~Hawking}

In \S\ref{sec:the-boundary}, we constructed a measure \prob on $\Schw'$, where $\Schw \ci \HE \ci \Schw'$ is a certain Gel'fand triple. In this section, we develop a different but analogous measure on the space of infinite paths in $\bd \Graph$. We carry out this construction for harmonic functions on $(\Graph, \cond)$ in \S\ref{sec:path-space-of-a-general-random-walk}, where the measure is defined in terms of transition probabilities $p(x,y) = \frac{\cond_{xy}}{\cond(x)}$ of the random walk, and the associated cylinder sets. When the random walk on $(\Graph, \cond)$ is transient, the current induced by a monopole gives a unit flow to infinity; such a current induces an orientation on the edges \edges and a new, naturally adapted, Markov chain. The state space of this new process is also \verts, but the transition probabilities are now defined by the induced current $p(x,y) = \frac{\curr(x,y)}{\act_\curr(x)}$. We call the fixed points of the corresponding transition operator the ``forward-harmonic'' functions, and carry out the analogous construction for them in \S\ref{sec:forward-harmonic-functions}. The authors are presently working to determine whether or not these measures can be readily related to each other or the measure \prob of \S\ref{sec:Gel'fand-triples-and-duality}. 


\section{The path space of a general random walk}
\label{sec:path-space-of-a-general-random-walk}

We begin by recalling some terms from \S\ref{sec:trace-resistance}, and providing some more detail. Let $\cpath = (x_0, x_1, x_2, \dots, x_n)$ be any finite path starting at $x=x_0$. The probability of a random walk started at $x$ traversing this path is
  \linenopax
  \begin{equation}\label{eqn:def:path-probability-recall}
    \prob(\cpath) := \prod_{k=1}^{n} p(x_{k-1},x_k),
  \end{equation}
  where $p(x,y) := \frac{\cond_{xy}}{\cond(x)}$ is the probability that the walk moves from $x$ to $y$ as in \eqref{eqn:def:p(x,y)}.
  \glossary{name={\cpath},description={path},sort=G}
  \glossary{name={$\prob^{(\cond)}$},description={a measure on the space of all paths in $G$},sort=Pc}
  \glossary{name={$p(x,y)$},description={transition probability of the random walk on the network: $\cond_{xy}/\cond(x)$},sort=prob}
  This intuitive notion can be extended via Kolmogorov consistency to the space of all infinite paths starting at $x$. Let $X_n(\cpath)$ denote the \nth coordinate of \cpath; one can think of \cpath as an event and $X_n$ as the random walk (a random variable), in which case 
  \linenopax
  \begin{equation}\label{eqn:def:X_n}
    X_n(\cpath) = \text{location of the random walk at time $n$.}
  \end{equation}

\begin{defn}\label{def:Paths}\label{def:prob^cond}
  Let \Paths denote the \emph{space of all infinite paths} \cpath in $(\Graph, \cond)$. Then a \emph{cylinder set} in \Paths is specified by fixing the first $n$ coordinates:
  \linenopax
  \begin{align}\label{eqn:def:cylinder-set-in-Paths(x)}
    \Paths_{(x_1,x_2,\dots,x_n)} := \{\cpath \in \Paths \suth X_k(\cpath) = x_k, k=1,\dots,n\}.
  \end{align}
  Define $\prob^{(\cond)}$ on cylinder sets by 
  \linenopax
  \begin{align}\label{eqn:def:prob^cond}
    \prob^{(\cond)}(\Paths_{(x_1,x_2,\dots,x_n)}) := \prod_{i=1}^n p(x_{i-1},x_i).
  \end{align}
\end{defn}

\begin{remark}
  It is clear from Definition~\ref{def:prob^cond} that the probability of a random walk following the finite path $\cpath = (x_0, x_1, x_2, \dots, x_n)$ is equal to the measure of the set of all infinite walks which agree with \cpath for the first $n$ steps: combining \eqref{eqn:def:path-probability-recall} and \eqref{eqn:def:prob^cond} gives $\prob^{(\cond)}(\Paths_{(x_1,x_2,\dots,x_n)}(x)) = \prob(\cpath)$. Observe that \eqref{eqn:def:prob^cond} is a conditional probability:
  \linenopax
  \begin{align}\label{eqn:cylinder-sets-as-conditional-prob}
    \prob^{(\cond)}(\Paths_{(x_1,x_2,\dots,x_n)}(x)) 
    = \prob^{(\cond)}\{\cpath \in \Paths(x) \,|\, X_k(\cpath) = x_k, k=1,\dots,n\}.
  \end{align}
\end{remark}


\begin{remark}[Kolmogorov consistency]
  \label{rem:Kolmogorov-consistency}
  We use Kolmogorov's consistency theorem to construct a measure on the space of paths beginning at vertex $x \in \verts$, see \cite[Lem.~2.5.1]{Jor06} for a precise statement of this extension principle in its function theoretic form and \cite[Exc.~2.4--2.5]{Jor06} for the method we follow here. The idea is that we consider a sequence of functionals $\{\gm^{(n)}\}$, where $\gm^{(n)}$ is defined on 
  \begin{equation}\label{eqn:def:An}
  \sA_n := \spn\{\charfn{\Paths_{(x_0,\dots,x_n)}} \suth x_i \nbr x_{i-1}, i=1,\dots,n\}.\end{equation}
Alternatively,
\begin{equation}\label{eqn:def:An-alt}
  \sA_n := \{f:\Paths \to \bR \suth f(\cpath_1)=f(\cpath_2) \text{ whenever } X_k(\cpath_1)=X_k(\cpath_2) \text{ for } k \leq n\}.
\end{equation}
That is, an element of $\sA_n$ cannot distinguish between two paths which agree for the first $n$ steps. This means that $\gm^{(n)}$ is a ``simple functional'' in the sense that it is constant on each cylinder set of level $n$:
\begin{equation}\label{eqn:def:An}
  \gm^{(n)}[f] = \sum_{x_0,\dots,x_n} a_{(x_0,\dots,x_n)} \gm^{(n)}[\charfn{\Paths_{(x_0,\dots,x_n)}}].
\end{equation}
If the functionals $\gm^{(n)}$ are mutually consistent in the sense that $\gm^{(n+1)}[f] = \gm^{(n)}[f]$, then Kolmogorov's consistency theorem gives a unique Borel probability measure on the space of all paths. More precisely, Kolmogorov's theorem gives the existence of a limit functional which is defined for functions on paths of infinite length, and this corresponds to a measure by Riesz's Theorem; see \cite{Jor06,Kol56}.
\end{remark}

In the following, we let \one denote the constant function with value equal to 1.

\begin{theorem}[Kolmogorov]\label{thm:Kolmogorov-extension-thm}
  If each $\gm^{(n)}:\sA_n \to \bC$ is a positive linear functional satisfying the consistency condition
  \begin{equation}\label{eqn:Kolmogorov-consistency}
    \gm^{(n+1)}[f] = \gm^{(n)}[f], \qq\text{for all } f \in \sA_n,
  \end{equation}
  then there exists a positive linear functional \gm defined on the space of functions on infinite paths such that
  \begin{equation}\label{eqn:Kolmogorov-functional}
    \gm[f] = \gm^{(n)}[f], \qq f \in \sA_n,
  \end{equation}
  where $f$ is considered as a function on an infinite path which is zero after the first
  $n$ edges. 
  Moreover, if we require the normalization $\gm^{(n)}[\one] = 1$, then \gm is determined uniquely.
\end{theorem} 

  We now show that $\prob^{(\cond)}$ extends to a natural probability measure on the space of infinite paths $\Paths(x)$.

\begin{theorem}\label{thm:P^cond-exists}
  For $(\Graph,\cond)$, there is a unique measure $\prob^{(\cond)}$ defined on $\Paths$ which satisfies
  \begin{equation}\label{eqn:P^cond-consistency}
    \bE[V] 
    = \int_{\Paths} V \,d\prob^{(\cond)} 
    = \int_{\Paths} V \,d\prob^{(\cond,n)} 
    = \bE^{(n)}[V], \qq \forall V \in \sA_n.
  \end{equation}
  \begin{proof}
    We must check condition \eqref{eqn:Kolmogorov-consistency} for $\gm^{(n)} = \prob^{(\cond,n)}$, defined by 
    \linenopax
    \begin{align*}
      \prob^{(\cond,n)}(\charfn{\Paths_{(x_0,\dots,x_n)}}) 
      := \prod_{i=1}^n p(x_{i-1},x_i)    
    \end{align*}
    with \eqref{eqn:def:prob^cond} in mind. 
    
    
    Think of $V \in \sA_n$ as an element of $\sA_{n+1}$ and apply $\prob^{(\cond,n+1)}$ to it:
    \linenopax
    \begin{align*}
      \prob^{(\cond,n+1)}[V]
      &= \sum_{x_0,\dots,x_{n+1}} a_{(x_0,\dots,x_{n+1})} 
        \prob^{(\cond,n+1)}(\charfn{\Paths_{(x_0,\dots,x_{n+1})}}) \\
      &= \sum_{x_0,\dots,x_n} \sum_{x_{n+1}} a_{(x_0,\dots,x_{n})} \prod_{i=1}^{n} p(x_{i-1},x_i)p(x_n,x_{n+1}) \\
      &= \sum_{x_0,\dots,x_n} a_{(x_0,\dots,x_{n})} \prod_{i=1}^{n} p(x_{i-1},x_i) \sum_{x_{n+1} \nbr x_n} p(x_n,x_{n+1}) \\
      &= \prob^{(\cond,n)} [V], \vstr[2.5]
      \qedhere
    \end{align*}
    since $\sum_{x_{n+1} \nbr x_n} p(x_n,x_{n+1}) = 1$. For the second equality, note that $f \in \sA_n$, so we can use the same constant $a$ for each $(n+2)$-tuple that begins with $(x_0,\dots,x_n)$. 
  \end{proof}
\end{theorem}

\subsection{A boundary representation for the bounded harmonic functions} 

\begin{defn}\label{def:bdd-cocycle}
  A \emph{cocycle} $V:\Paths \to \bR$ is a measurable function on the infinite path space which is independent of the first finitely many vertices in the path:
  \begin{equation}\label{eqn:def:bdd-cocycle-condition}
    V(\cpath) = V(\shift\cpath),
  \end{equation}
  where \shift is the shift operator, i.e., if $\cpath=(x_0,x_1,x_2,\dots)$, then $\shift\cpath=(x_1,x_2,x_3,\dots)$.
\end{defn}

Intuitively, a cocycle is a function on the boundary $\bd \Graph$; it depends only on the asymptotic trajectory of a path/random walk. A cocycle does not care where the random walk began, only where it goes.
\version{}{\marginpar{Make this precise}}
More precisely, a cocycle is a special kind of martingale, as we will see below. 

The goal of this section is to show that the bounded harmonic functions are in bijective correspondence with the cocycles; see Theorem~\ref{thm:harmonic-functions=cocycles}. That is, the formula 
  \begin{equation}\label{eqn:def:harmonic-vs-cocycle}
    h(x) = \Ex_x[V]Ê
  \end{equation}
spells out a bijective correspondence between functions $h \in \Harm$, and cocycles $V$ on the space of infinite paths. Our present concern is the space of all bounded harmonic functions; we will presently consider the class of finite-energy functions. A good reference for this section is \cite[Thm.~2.7.1]{Jor06}.

Note that the left hand side of \eqref{eqn:def:harmonic-vs-cocycle} involves no measure theory, in contrast to the right-hand side, where the expectation refers to the integration of cocycles $V$ against the probability measure $\prob^{(\cond)}$. The underlying Borel probability space of $\prob^{(\cond)}$ is the \gs-algebra of measurable sets generated by the cylinder sets in \Paths, i.e., by the subsets in \Paths which fix only a finite number of places (in the infinite paths). 

The condition on a measurable function $V$ on \Paths which accounts for $h$ defined by \eqref{eqn:def:harmonic-vs-cocycle} being harmonic is that $V$ is invariant under a finite left shift; cf.~\eqref{eqn:def:bdd-cocycle-condition}. It turns out that in making the integrals $\Ex_x(V)$ precise, the requirement that $V$ be measurable is a critical assumption. In fact, there is a variety of non-measurable candidates for such functions $V$ on \Paths.

\begin{defn}\label{def:path-expectation}
  For any measurable function $V:\Paths \to \bR$, we write 
  \begin{equation}\label{eqn:def:path-expectation}
    \bE_x[V] := \bE[V \,|\, X_0=x] = \int_{\Paths(x)} V(\cpath) \,d\prob^{(\cond)}
  \end{equation}
  for the expected value of $V$, conditioned on the path starting at $x$.
\end{defn}

\begin{lemma}\label{thm:harmonic-as-integral-identity}
  For $h \in \Harm$ and any $n=1,2,\dots,$
  \linenopax
  \begin{align}\label{eqn:int(h(Xn)=h}
    h(x) = \int_{\Paths_x} h \comp X_n \, d\prob^{(\cond)}.
  \end{align}
  \begin{proof}  
    By the definition of the cylinder measure $d\prob^{(\cond)}$ (Definition~\ref{def:prob^cond}),
    \linenopax
    \begin{align}\label{eqn:int(h(Xn)=h}
      \int_{\Paths_x} h \comp X_1 \, d\prob^{(\cond)}
      = \sum_{y \nbr x} p(x,y) \int_{\Paths_y} h \comp X_0 \, d\prob^{(\cond)}
      = \sum_{y \nbr x} p(x,y) h(y) \int_{\Paths_y} \, d\prob^{(\cond)}
      = \Prob h(x),
    \end{align}
    so iteration and $\Prob h = h$ gives $\int_{\Paths_x} h \comp X_n \, d\prob^{(\cond)} = h(x)$ for every $n=1,2,\dots$.
  \end{proof} 
\end{lemma}
    
\begin{theorem}\label{thm:harmonic-functions=cocycles}
  The bounded harmonic functions are in bijective correspondence with the cocycles. More precisely, if $V$ is a cocycle, then it defines a harmonic function via
  \linenopax
  \begin{equation}\label{eqn:hV=E[V]}
    h_V(x) := \bE_x[V].
  \end{equation}
  Conversely, if $h$ is harmonic, then it defines a cocycle via
  \linenopax
  \begin{equation}\label{eqn:Vh-is-martingale-limit}
    V_h(\cpath) := \lim_{n \to \iy} h(X_n(\cpath)), \qq \text{for } \prob^{(\cond)}-\text{a.e. } \cpath \in \Paths(x).
  \end{equation}
  \begin{proof}
    \fwd Recall that $\Lap = \cond - \Trans$; we will show that $\cond h_V = \Trans h_V$ whenever $V$ is a cocycle. If $\Paths_{(x,y)} := \{\cpath \in \Paths(x) \suth X_1(\cpath) = y\}$, then $\Paths(x) = \bigcup_{y \nbr x} \Paths_{(x,y)}$ is a disjoint union and
    \linenopax
    \begin{align*}
     h_V(x) 
     = \bE_x[V] 
      = \int_{\Paths(x)} V(\cpath) \,d\prob^{(\cond)} 
      = \sum_{y \nbr x} \int_{\Paths_{(x,y)}} V(\cpath) \,d\prob^{(\cond)}.
    \end{align*}
    For each $\cpath \in \Paths_{(x,y)}$, one has $\prob(\cpath) = \prob(x,y) \prob(\shift \cpath) = p(x,y) \prob(\shift \cpath)$ by \eqref{eqn:def:path-probability-recall}, whence
    \linenopax
    \begin{align*}
      \cond(x) h_V(x) 
      
      &= \cond(x) \sum_{y \nbr x}\int_{\Paths_{(x,y)}} \negsp[15] p(x,y) V(\shift\cpath) \,d\prob^{(\cond)} 
      = \sum_{y \nbr x} \cond_{xy} \int_{\Paths(y)} V(\cpath) \,d\prob^{(\cond)} 
     = \Trans h_V(x), 
    \end{align*}
    where the cocycle property \eqref{eqn:def:cocycle-condition} is used for the second equality.
    
    \bwd Now let $h$ be a bounded harmonic function. Since 
    \linenopax
    \begin{align*}
      \lim_{n \to \iy} h(X_n(\cpath))
      = \lim_{n \to \iy} h(X_{n+1}(\cpath))
      = \lim_{n \to \iy} h(X_n(\shift\cpath)), 
    \end{align*}
    the cocycle property \eqref{eqn:def:cocycle-condition} is obviously satisfied whenever the limit exists. Let $\gS_n$ denote the \gs-algebra generated by the cylinder sets of level $n$, and denote $X_n(\cpath) = x_n$. Then $X_{n+1}(\cpath)$ is a neighbour $y \nbr x_n$, and 
   \linenopax
    \begin{align}
      \bE[ h(X_{n+1}) \,|\, \gS_n]
      &= \bE[ h(X_{n+1}) \,|\, \gS_n] \sum_{y \nbr x_n} p(x_n,y) \notag \\
      &= \sum_{y \nbr x_n} p(x_n,y) \bE[ h(X_{n+1}) \,|\, \gS_n] \notag \\
      &= \bE\left[ \sum_{y \nbr x_n} p(x_n,y) h(X_{n+1}) \,|\, \gS_n\right] \notag \\
      &= \bE[h(X_n) \,|\, \gS_n] \vstr \notag \\
      &= h(X_n). \vstr 
    \end{align}
    Since $h$ is bounded, this shows $h(X_n)$ is a bounded martingale, whence by Doob's Theorem (cf.~\cite{Doob53}), it converges pointwise $\prob^{(\cond)}$-a.e. on \paths and \eqref{eqn:Vh-is-martingale-limit} makes $V_h$ well-defined $\prob^{(\cond)}$-a.e. on \paths.
    
    ($\leftrightarrow$) We conclude with a proof that these two constructions correspond to inverse operations. 
    If $V$ is a cocycle, we must show that $\lim_{n \to \iy} \Ex_{X_n(\cpath)}[V] = V(\cpath)$. To this end, for $A \ci \Paths$, define the conditioned measure $\prob_A := \frac{\prob^{(\cond)}(A \cap \cdot )}{\prob^{(\cond)}(A)}$, so that $d\prob_A = \frac1{\prob^{(\cond)}(A)} \charfn{A} d\prob^{(\cond)}$. Now for a fixed $\cpath \in \Paths$, let $A_n = \Paths_{(x,X_1(\cpath),\dots,X_n(\cpath))}$ be the cylinder set whose first $n+1$ coordinate agree with \cpath. Applying the measure identity $\lim \gm(A_n) = \bigcap \gm(A_n)$ for nested sets, we obtain $\lim_{n \to \iy} \prob_{A_n} = \gd_\cpath$ as a weak limit of measures. Now
    \linenopax
    \begin{align}\label{eqn:Ex{Xn}[V]=V:derivation1}
      \Ex_{X_n(\cpath)}[V]
      = \int_{\Paths_{X_n(\cpath)}} V(\gx) \, d\prob^{(\cond)}(\gx)
      = \int_{\Paths} V(\gx) \, d\prob_{A_n}(\gx)
      \limas{n} \int_{\Paths} V(\gx) \, d\gd_\cpath
      = V(\cpath).
    \end{align} 
    On the other hand, if $h$ is harmonic, we must show $\Ex_x[V_h]=h$. Then for $V_h(\cpath) := \lim_{n \to \iy} h(X_n(\cpath))$, boundedness allows us to apply the dominated convergence theorem and compute
    \linenopax
    \begin{align}\label{eqn:Ex[Vh]=h}
     \Ex_x[V_h]
     = \int_{\Paths_x} \lim_{n \to \iy} \left(h \comp X_n(\cpath)\right) \,d\prob^{(\cond)}
     = \lim_{n \to \iy} \int_{\Paths_x} h \comp X_n(\cpath) \,d\prob^{(\cond)}.
    \end{align} 
    Now the sequence on the right-hand side of \eqref{eqn:Ex[Vh]=h} is constant by Lemma~\ref{thm:harmonic-as-integral-identity}, so $\Ex_x[V_h] = h(x)$.
  \end{proof}
\end{theorem}

\begin{remark}
  The \fwd direction of the proof of Theorem~\ref{thm:harmonic-functions=cocycles} may also be computed 
  \linenopax
  \begin{align*}
    h_V(x) = \cond(x) \bE_x[V] 
    &= \cond(x) \sum_{y \nbr x} p(x,y) \bE_x[V \,|\, X_1=y]  
    = \sum_{y \nbr x} \cond_{xy} \bE_y[V] 
    = \sum_{y \nbr x} \cond_{xy} h_V(y),
  \end{align*}
  where $\bE_x[V \,|\, X_1=y] = \bE_y[V]$ because the random walk is a Markov process.   See, e.g., \cite[Prop.~9.1]{LevPerWil08}.
\end{remark}

\version{}{
\begin{defn}\label{def:space-of-bounded-paths}
  Let $\gQ_n := \{\cpath \in \Paths \suth |X_k(\cpath)| \leq n, \forall k\}$, where $|x|$ is the shortest-path distance from $o$ to $x$. Then $\gQ := \bigcup_n \gQ_n$ is the collection of all \emph{bounded paths} in \Graph.
\end{defn}

\begin{cor}\label{thm:bdd-paths-have-measure-0}
  If $\Harm \neq 0$, then $\prob^{(\cond)}(\gQ) = 0$.
  \begin{proof}
    \version{}{\marginpar{This doesn't quite work.}}
    Let $h \in \Harm$ be nonconstant. Since $X_k(\cpath) \neq X_{k+1}(\cpath)$ for any \cpath, the limit $\lim_{n \to \iy} h(X_n(\cpath)) = V(\cpath)$ cannot exist for any $\cpath \in \gQ$. The conclusion follows by \eqref{eqn:Vh-is-martingale-limit}. 
  \end{proof}
\end{cor}
}


\section{The forward-harmonic functions}
\label{sec:forward-harmonic-functions}

The current passing through a given edge may be interpreted as the expected value of the number of times that a given unit of charge passes through it. This perspective is studied extensively in the literature; see \cite{DoSn84,Lyons:ProbOnTrees} for excellent treatments. In this case, $p(x,y) = \frac{\cond_{xy}}{\cond(x)}$ helps one construct a current which is harmonic, or dissipation-minimizing. However, that is not what we do here; we are interested in studying current functions whose dissipation is finite but not necessarily minimal. In Theorem~\ref{thm:Paths-is-nonempty}, we show that the experiment always induces a ``downstream'' current flow between the selected two points; that is, a path along which the potential is strictly decreasing.

These probability notions will play a key role in our solution to questions about \emph{activity}; cf.~Definition~\ref{def:activity}. We use the forward path measure again in our representation formula (Theorem~\ref{thm:forward-harmonic-functions=cocycles}) for the class of forward-harmonic functions on \Graph. The corresponding Markov process is dynamically adapted to the network (and the charge on it). This representation is dynamic and nonisotropic, which sets it apart from other related representation formulas in the literature.

\subsection{Activity of a current and the probability of a path}
\label{sec:current-paths}
Given a (fixed) current, we are interested in computing ``how much of the current'' takes any specified path from $x$ to some other (possibly distant) vertex $y$. This will allow us to answer certain existence questions (see Theorem~\ref{thm:Paths-is-nonempty}) and provides the basis for the study of the forward-harmonic functions studied in \S\ref{sec:forward-harmonic-functions}. Note that, in contrast to \eqref{eqn:def:path-probability-recall}, the probabilistic interpretation given in Definition~\ref{def:path-probability} (and the discussion preceding it) does not make any reference to \cond. In this section we follow \cite{Pow76b} closely.

\begin{defn}\label{def:activity}
  The \emph{divergence} of a current $\curr:\edges \to \bR$ is the function on $x \in \verts$ defined by
  
  
  
  
  
  \begin{equation}\label{eqn:def:absolute-activity}
    \act_\curr(x) :=
    \begin{cases}
      \frac12 \sum_{y \nbr x} |\curr(x,y)|, &x \neq \ga,\gw \\
      1, &x=\ga,\gw.
    \end{cases}
  \end{equation}
  which describes the total ``current traffic'' passing through $x \in \verts$.
  Thus, \act is an operator mapping functions on \edges to functions on \verts; see \S\ref{sec:divergence-operator} for details.
\end{defn}

For convenience, we restate Definition~\ref{def:current-path-preview}:

\begin{defn}\label{def:current-path}
  Let $v:\verts \to \bR$ be given and suppose we fix \ga and \gw for which $v(\ga) > v(\gw)$. Then a \emph{current path} \cpath (or simply, a \emph{path}) is an edge path from \ga to \gw with the extra stipulation that $v(x_k) < v(x_{k-1})$ for each $k=1,2,\dots,n$.
  Denote the set of all current paths by $\Paths = \Paths_{\ga,\gw}$ (dependence on the initial and terminal vertices is suppressed when context precludes confusion). Also, define $\Paths_{\ga,\gw}(x,y)$ to be the subset of current paths from \ga to \gw which pass through the edge $(x,y) \in \edges$.
  \glossary{name={\Paths},description={set of paths},sort=G,format=textbf}
\end{defn}


      \begin{comment}\label{rem:not-every-flow-has-paths}
        It is clear for an arbitrary function $v: \verts \to \bR$ and $\ga,\gw \in \verts$, it may be the case that $\Paths_{\ga,\gw} = \es$. We will soon show that if $v$ is a potential function (see Definition~\ref{def:dipole}), then there is always a induced current flow. However, it is a somewhat surprising fact this flow may not contain any current paths! See the one-sided ladder model described in Example~\ref{exm:one-sided-ladder-model} for a potential function whose induced flow contains no current paths. As a consequence, the hypothesis $\energy(v) < \iy$ can not be omitted from the statement of Lemma~\ref{thm:Paths-is-nonempty}.
      \end{comment}

Suppose we fix a source \ga and sink \gw and consider a single current path \cpath from \ga to \gw. With $\act_\curr$ defined as in \eqref{eqn:def:absolute-activity}, one
can consider $\frac{\curr(x,y)}{\act_\curr(x)}$ as the probability that a unit of charge at $x$ will pass to a ``downstream'' neighbour $y$. Note that $\curr(x,y) > 0$ and $\act_\curr \neq 0$, since we are considering an edge of our path \cpath. This allows us to define a probability measure on the path space $\Paths_{\ga,\gw}$.

\begin{defn}\label{def:path-probability}
  If $\cpath \in \Paths$ follows the vertex path $(\ga = x_0, x_1, x_2, \dots, x_n = \gw)$, the define the probability of \cpath by
  \begin{equation}\label{eqn:def:path-probability}
    \prob(\cpath) := \prod_{k=1}^{n} \frac{\curr(x_{k-1},x_k)}{\act_\curr(x_{k-1})}.
  \end{equation}
  This quantity gives the probability that a unit of charge at \ga will pass to \gw by traversing the path \cpath.
\end{defn}


\version{}{
The following result fills another hole in the proof of $\dist_\Lap(x,y) = \dist_\diss(x,y)$ (recall \eqref{eqn:def:R(x,y)-Lap} and \eqref{eqn:def:R(x,y)-diss}) in \cite{Pow76b}.

\begin{cor}\label{thm:no-infinite-current-paths-for-minimal-currents}
  \marginpar{Is this even true?}
  If \curr minimizes \diss on $\Flo(\ga,\gw)$, then all current paths flow from \ga to \gw.
  \begin{proof}
    If $\curr \in \Flo(\ga,\gw)$ is minimal, then Theorem~\ref{thm:minimal-flows-are-induced-by-minimizers} implies \curr is induced by a potential $v$ which minimizes \energy over $\Pot(\ga,\gw)$. Since orthogonality gives $\energy(v) = \energy(\Pfin v) + \energy(\Phar v)$, such a minimal $v$ must lie in \Fin. In fact, $v = f_\ga - f_\gw$, where $f_x = \Pfin v_x$. 
  \end{proof}
\end{cor}
}


\subsection{Forward-harmonic transfer operator}
\label{sec:forward-harmonic-transfer-operator}

In this section we consider the functions $h : \verts \to \bC$ which are \emph{forward-harmonic}, that is, functions which are harmonic with respect to a current \curr. We make the standing assumption that the network is transient; this guarantees the existence of a monopole at every vertex, and the induced current will be a unit flow to infinity; cf.~Corollary~\ref{thm:transient-here-implies-transient-everywhere}.

We orient the edges by a fixed unit current flow \curr to infinity, as in Definition~\ref{def:orientation}. The forward-harmonic functions functions are fixed points of a transfer operator induced by the flow which gives the value of $h$ at one vertex as a convex combination of its values at its downstream neighbours.

The main idea is to construct a measure on the space of paths beginning at vertex $x \in \verts$, and then use this measure to define forward-harmonic functions. In fact, we are able to produce all forward-harmonic functions from the class of functions which satisfy a certain cocycle condition, see Definition~\ref{def:cocycle}.

In Theorem~\ref{thm:forward-harmonic-functions=cocycles} we give an integral representation for the harmonic functions, and in Corollary~\ref{thm:current-with-sink=>trivial-harmonic-functions} we show that if \curr has a universal sink, then the only forward harmonic functions are the constants.

\begin{remark}[A current induces a direction on the resistance network.]
  If we fix a minimal current $\curr = \Pdrp \curr$, the flow gives a strict partial order on \verts and the flags in the resulting poset are the induced current paths. Thus we say $x \prec y$ iff $x$ is upstream from $y$, that is, iff there exists a current path from $x$ to $y$ in the sense of Definition~\ref{def:current-path}. Since \curr is minimal, $x \prec y$ implies $y \nprec x$ and $x \nprec x$. Transitivity is immediate upon considering the concatenation of two finite paths.
\end{remark}

\begin{defn}\label{def:set-of-current-paths}
  Given a fixed minimal current $\curr = \Pdrp \curr$, we denote the \emph{set of all
  current paths} in the resistance network $(\Graph,\cond)$ by
  \begin{equation}\label{eqn:def:set-of-current-paths}
    \paths_\curr := \{\cpath = (x_0,x_1, \dots \suth (x_i,x_{i+1}) \in \edges, x_i \prec x_{i+1}\}.
  \end{equation}
  For $n = 1, 2, \dots$ , we denote the set of all current paths of length n by
  \begin{equation}\label{eqn:def:set-of-current-paths-length-n}
    \paths^{(n)}_\curr := \{\cpath = (x_0, x_1, \dots, x_n) \suth (x_i,x_{i+1}) \in \edges, x_i \prec x_{i+1}\},
  \end{equation}
  and denote the collection of paths starting at $x$ by $\paths_\curr(x) := \{\cpath \in \paths_\curr \suth x_0=x\}$, and likewise for $\paths^{(n)}_\curr(x)$.
\end{defn}


Here, the orientation is determined by \curr, and if $\curr(x,y) = 0$ for some $(x,y) \in \edges$, then this edge will not appear in any current path, and for all practical purposes it may be considered as having been removed from \Graph for the moment.

\begin{defn}\label{def:forward-neighbours}
  When a minimal current $\curr = \Pdrp \curr$ is fixed, the set of \emph{forward neighbours} of $x \in \verts$ is
  \begin{equation}\label{eqn:def:forward-neighbours}
    \fnbrs_\curr(x) := \{y \in \verts \suth x \prec y, x \nbr y\}.
  \end{equation}
\end{defn}

\begin{defn}\label{def:forward-Laplacian}
  If $v:\verts \to \bR$, define the \emph{forward Laplacian} of $v$ by
  \begin{equation}\label{eqn:def:forward-Laplacian}
    \fLap v(x) := \sum_{y \in \fnbrs(x)} \cond_{xy}(v(x) - v(y))
  \end{equation}
  A function $h$ is forward-harmonic iff $\fLap h = 0$.
\end{defn}

For Definitions~\ref{def:set-of-current-paths}--\ref{def:forward-Laplacian}, the dependence on \curr may be suppressed when context precludes confusion. 

\begin{theorem}\label{thm:Px-exists}
  For $\curr \in H_\diss$ and $x \in \verts$, there is a unique measure $\prob_x$ defined on $\paths_\curr(x)$ which satisfies
  \begin{equation}\label{eqn:Px-consistency}
    \prob_x[f] = \prob_x^{(n)}[f], \qq f \in \sA_n.
  \end{equation}
  \begin{proof}
    We only need to check Kolmogorov's consistency condition \eqref{eqn:Kolmogorov-consistency}; see \cite{Jor06,Kol56}. For $n < m$, consider $\sA_n \ci \sA_m$ by assuming that $f$ depends only on the first $n$ edges of \cpath. (For brevity, we denote a function on $n$ edges as a function on $n+1$ vertices.) Then
    \linenopax
    \begin{align*}
      \prob_x^{(m)}[f]
      &= \int_{\paths_\curr(x)} f(\cpath) \, d\prob_x^{(m)}(\cpath) \\
      &= \int_{\paths_\curr(x)} f(x_0,x_1,x_2,\dots,x_n) \, d\prob_x^{(m)}(\cpath) \\
      &= \int_{\paths_\curr(x)} f(x_0,x_1,x_2,\dots,x_n) \, d\prob_x^{(n)}(\cpath) \\
      &= \prob_x^{(n)} [f]. \vstr[2.5]
      \qedhere
    \end{align*}
  \end{proof}
\end{theorem}

\subsection{A boundary representation for the forward-harmonic functions} We now show that the forward-harmonic functions are in bijective correspondence with the cocycles, when defined as follows.

\begin{defn}\label{def:cocycle}
  A \emph{cocycle} is a function $f:\paths_\curr \to \bR$ which is compatible with the probabilities on current paths in the sense that it satisfies
  \begin{equation}\label{eqn:def:cocycle-condition}
    f(\cpath) = f(x_0,x_1,x_2,x_3\dots) = \frac{\cond_{x_0x_1} \act_\curr(x_0)}{\cond^+(x_0) \curr(x_0,x_1)} f(x_1,x_2,x_3\dots),
  \end{equation}
  whenever $\cpath=(x_0,x_1,x_2,x_3\dots) \in \paths_\curr$ is a current path as in Definition~\ref{def:set-of-current-paths}, and $(x_0,x_1)$ is the first edge in \cpath. Also, $\cond^+(x) := \sum_{y \in \fnbrs(x)} \cond_{xy}$ is the sum of conductances of edges \emph{leaving} $x$. If the operator $m$ is given by multiplication by
  \begin{equation}\label{eqn:def:m-multiplication-operator}
    m(x,y) = \frac{\cond_{xy} \act_\curr(x)}{\cond^+(x) \curr(x,y)},
  \end{equation}
  and \gs denotes the shift operator, the cocycle condition can be rewritten $f = mf\gs$. Using $e_k = (x_{k-1}, x_k)$ to denote the edges, this gives
  \begin{equation}\label{eqn:f-as-infinite-product}
    f(e_1, e_2, \dots)
    = m(e_1) \dots m(e_n) f(e_{n+1}, e_{n+2}, \dots)
    = \prod_{k=1}^\iy m(e_k).
  \end{equation}
\end{defn}

\begin{defn}\label{def:forward-transfer-operator}
  Define the \emph{forward transfer operator} $\Trans_\curr$ induced by \curr by
  \begin{equation}\label{eqn:def:forward-transfer-operator}
    (\Trans_\curr f)(x) := \frac1{\cond^+(x)} \sum_{y \in \fnbrs(x)} \cond_{xy} f(y).
  \end{equation}
\end{defn}

\begin{lemma}\label{thm:forward-cocycles-are-harmonic}
  If $f:\paths(x) \to \bR$ is a cocycle and one defines $h_f(x) := \prob_x[f]$, it
  follows that $h_f(x)$ is a fixed point of the forward transfer operator $\Trans_\curr$.
  \begin{proof}
    With $h_f$ so defined, we conflate the linear functional $\prob_x$ with the measure associated to it via Riesz's Theorem and compute
    \linenopax
    \begin{align*}
      (\Trans_\curr h_f)(x)
      &= \frac1{\cond^+(x)} \sum_{y \in \fnbrs(x)} \cond_{xy} \prob_y[f]
        &&\text{def } \Trans, h_f \\
      &= \sum_{y \in \fnbrs(x)} \frac{\cond_{xy}}{C^+(x)}
        \int_{\paths_\curr(y)} f(\cpath) \,d\prob_y(\cpath)
        &&\prob_y \text{ as a measure} \\
      &= \sum_{y \in \fnbrs(x)} \int_{\paths_\curr(x)}
        \frac{\cond_{xy}}{\cond^+(x)} f(\gs\cpath) \,d\prob_x(\cpath)
        &&\text{change of vars} \\
      &= \sum_{y \in \fnbrs(x)} \frac{\curr(x,y)}{\act_\curr(x)} \int_{\paths_\curr(x)}
        \frac{\cond_{xy}}{\cond^+(x)} \frac{\act_\curr(x)}{\curr(x,y)} f(\gs\cpath) \,d\prob_x(\cpath)
        &&\text{just algebra} \\
      &= \sum_{y \in \fnbrs(x)} \frac{\curr(x,y)}{\act_\curr(x)}
        \int_{\paths_\curr(x)} f(\cpath) \,d\prob_x(\cpath)
        &&\text{by \eqref{eqn:def:cocycle-condition}} \\
      &= \sum_{y \in \fnbrs(x)} \frac{\curr(x,y)}{\act_\curr(x)} \prob_x(f)
        &&\prob_x \text{ as a functional} \\
      &= \prob_x(f) \vstr[2.5]
        &&\textstyle{\sum \tfrac{\curr(x,y)}{\act_\curr(x)} = 1}
    \end{align*}
    To justify the change of variables, note that if \cpath is a path starting at $x$ whose  first edge is $(x,y)$, then \gs\cpath is a path starting at $y$. Moreover, since $y$ is a downstream neighbour of $x$, every path \cpath starting at $y$ corresponds to exactly one path starting at $x$, namely, $((x,y), \cpath)$.
  \end{proof}
\end{lemma}

\begin{theorem}\label{thm:forward-harmonic-functions=cocycles}
  The forward-harmonic functions are in bijective correspondence with the cocycles. More precisely, if $f$ is a cocycle, then
  \begin{equation}\label{eqn:hf-is-P[f]}
    h_f(x) := \prob_x[f]
  \end{equation}
  is harmonic. Conversely, if $h$ is harmonic, then
  \begin{equation}\label{eqn:fh-is-martingale-limit}
    f_h(\cpath) := lim_{n \to \iy} h(X_n(\cpath)), \qq \cpath \in \paths_x
  \end{equation}
  is a cocycle, where $X_n(\cpath)$ is the \nth vertex from $x$ along the path \cpath.
  \begin{proof}
    \fwd Let $f$ be a cocycle and define $h_f$ as in \eqref{eqn:hf-is-P[f]} with $C^+(x)$ as in Definition~\ref{def:forward-transfer-operator}, compute
    \linenopax
    \begin{align*}
      \fLap h_f(x)
      &= \sum_{y \in \fnbrs(x)} \cond_{xy} (\prob_x[f] - \prob_y[f]) \\
      &= \prob_x[f] \sum_{y \in \fnbrs(x)} \cond_{xy}
        - \sum_{y \in \fnbrs(x)} \cond_{xy} \prob_y[f] \\
      &= \prob_x[f] C^+(x) - \sum_{y \in \fnbrs(x)} \cond_{xy} \prob_y[f],
    \end{align*}
    which is 0 by Lemma~\ref{thm:forward-cocycles-are-harmonic}.

    \bwd Let $h$ satisfy $\fLap h = 0$. Observe that $X_n$ is a Markov chain with transition probability $\prob_x$ at $x$. The above computations show that $h$ is then a fixed point of $\Trans_\curr$, and hence $h(X_n)$ is a bounded martingale. By Doob's Theorem (cf.~\cite{Doob53}), it converges pointwise $\prob_x$-ae on \paths and \eqref{eqn:fh-is-martingale-limit} makes $f_h$ well-defined.
    One can see that $f_h$ is a cocycle by the same arguments as in the proof of \cite[Thm.~2.7.1]{Jor06}.
  \end{proof}
\end{theorem}


\begin{cor}\label{thm:current-with-sink=>trivial-harmonic-functions}
\version{}{\marginpar{If the support of $\prob_x$ is not all of \edges, then can two distinct cocycles give the same harmonic function?}}
  If \curr has a universal sink, in other words, if all current paths \cpath end at some common point \gw, then the only forward-harmonic function is the zero function.
  \begin{proof}
    Every harmonic function comes from a cocycle, which in turn comes from a harmonic function as a martingale limit, by the previous theorem. However, formula \eqref{eqn:fh-is-martingale-limit} yields
    \linenopax
    \begin{align*}
      f_h(\cpath) := \lim_{n \to \iy} h(X_n(\cpath)) = h(\gw), \qq \forall \cpath \in \paths.
    \end{align*}
    Thus every cocycle is constant, and hence \eqref{eqn:def:cocycle-condition} implies $f_h \equiv 0$. Then \eqref{eqn:hf-is-P[f]} gives $h \equiv 0$.
  \end{proof}
\end{cor}

\version{}{\marginpar{Actually, maybe \eqref{eqn:def:cocycle-condition} doesn't imply 0 ... without further restrictions on the network or something.}}


\section{Remarks and references}
\label{sec:Remarks-and-References-probab-interp}

The probability literature is among the largest of all the subdisciplines of mathematics, and so the following list of suggested references barely begins to scratch the surface: \cite{Ign08, AHKL06, Aik05, BrWo05, ChKi04, BuZo09, IKW09, Si09, YiZh05}.

Of these references, some are more specialized. However for prerequisite material (if needed), the reader may find the following sources especially relevant: \cite{Lyons:ProbOnTrees, LevPerWil08, AlFi09, Peres99}.
{"config": "arxiv", "file": "0806.3881/probab-interp.tex"}