While reading up on Markov Chains/Processes on graphs I came across the definition of a Green's function for a random walk on a network graph:
Let $(V, E)$ be a network graph. $Z \subset V, a \in V \setminus Z$
Then a Green's function $G_Z(x, a)$ is the expected number of visits to $x$ from a source $a$ before hitting the sink set $Z$.
To me this seems, slightly unrelated to the Green's function of a linear PDE (as response to forcing by Dirac's delta function) and I'm not sure how to connect my understanding.
For example, if we take the Green's function of the continuous heat equation, it's clear that the source set $a$ is now the origin but to me it's not clear what the set $Z$ maps to and how the process can be interpreted as a unit current flow. Is $G$ here still connected to some 'relative hit rate'?