Non-equilibrium steady state of the symmetric exclusion process with reservoirs

Consider the open symmetric exclusion process on a connected graph with vertexes in $[N-1]:=\{1,\ldots, N-1\}$ where points $1$ and $N-1$ are connected, respectively, to a left reservoir and a right reservoir with densities $\rho_L,\rho_R\in(0,1)$. We prove that the non-equilibrium steady state of such system is $$\mu_{\text{stat}} = \sum_{I\subset \mathcal P([N-1]) }F(I)\bigg(\otimes_{x\in I}\rm{Bernoulli}(\rho_R)\otimes_{y\in [N-1]\setminus I}\rm{Bernoulli}(\rho_L) \bigg).$$ In the formula above $\mathcal P([N-1])$ denotes the power set of $[N-1]$ while the numbers $F(I)> 0$ are such that $\sum_{I\subset \mathcal P([N-1]) }F(I)=1$ and given in terms of absorption probabilities of the absorbing stochastic dual process. Via probabilistic arguments we compute explicitly the factors $F(I)$ when the graph is a homogeneous segment.