Let $V = \{1,2,3,...,N\}$ be the vertex set of a graph. Let $d(i,j)>=0$ represent the $(i,j)$ vertex distance between vertices $i$ and $j$, $i \in V, j \in V$. Now, define a non-negative number $w_{i,j}$, for each edge $(i,j)$ of the graph, representing the probability of going to vertex $j$, given that we are now on vertex $i$. Therefore, $\sum_{j=1}^N w_{i,j}=1$, for all $i \in V$. Assume that it is impossible to leave vertex $N$, that is, $w_{N,j}=0$, for all $j \in V$, and assume that vertex $N$ is reachable from any vertex (there is a path starting at any vertex to vertex $N$ with non-zero probability). Moreover: $d(i,i)=0$, for all $i \in V$.
Finally, define a sequence of random variables, $(v_0, v_1, v_2, ..., v_t, ...)$, where $v_t$ represents the vertex we are in time $t$, $v_0$ is known (therefore, deterministic). It is known that $P(v_{t+1}=j; v_t=i)=w_{i,j}$, where $P(a;b)$ denotes the conditional probability of event $a$ given that event $b$ happened.
My goal is to find a closed form expression for the expected value of the total distance traversed, when moving from $v_0=m \in V$ to vertex $N$, $\mathbf E[\sum_{t=0}^{\infty} d(v_t,v_{t+1})]$. Could anyone give me a clue where to start?
Please, if problem is not well defined, feel free to ask.