A Jarzynski-type equality for nonequilibrium steady-state probabilities

We show that steady-state probabilities of a nonequilibrium Markovian system can be reconstructed from a weighted ensemble average of finite-time loop-erased paths. Each path $\mathrm{\Gamma }$ is weighted by $e^{-S(\mathrm{\Gamma })}$, where $S(\mathrm{\Gamma })$ can be interpreted either as the action functional or as the entropy change in the surrounding reservoirs. In doing so, we uncover a Jarzynski-type equality for nonequilibrium steady states. We also reveal that the probabilities of loop-erased paths obey strong thermodynamic symmetry relations.