Green's function of a Markov process

Summary

The Green's function of a Markov process is the density of its potential kernel with respect to a reference measure. Its value $u(x, y)$ measures how much time, on average, the process started at $x$ spends near $y$. For Brownian motion it is a constant multiple of the classical Green's function of the Laplacian. For a Markov chain it is the expected number of visits.