Given a resistive electrical circuit $G$, i.e. a graph with nonzero weights attached to each edge whose reciprocal we call the "resistance," we can define a reversible markov chain on the graph, provided that the graph is at most countable and the weights are summable at each vertex. One then defines the notion of a voltage between two disjoint subsets $A$ and $Z$ of the graph to be any function that is harmonic on $G-(A\cup Z)$, where the harmonicity is meant relative to this derived MC. This suggests to me that perhaps in general every solution to the Dirichlet problem for any "nice" subset of a state space of a Feller Process or MC is also a potential for some vector field that one can naturally interpret in physical terms. For instance, in the graph theoretic example we discussed, the vector field is the electric field induced by the voltage difference in the circuit. Is this true in general? What is the associated vector field of interest in the case of Brownian Motion?
By the way, the interested reader can consult the same resource as I did to get this far: