  For boundary-driven non-equilibrium Markov models of non-interacting
particles in one dimension, either in continuous space with the Fokker-Planck
dynamics involving an arbitrary force $F(x)$ and an arbitrary diffusion
coefficient $D(x)$, or in discrete space with the Markov jump dynamics
involving arbitrary nearest-neighbor transition rates $w(x \pm 1,x)$, the
Markov generator can be transformed via an appropriate similarity
transformation into a quantum supersymmetric Hamiltonian with many remarkable
properties. In particular, the mapping from the boundary-driven non-equilibrium
dynamics towards some dual equilibrium dynamics [J. Tailleur, J. Kurchan and V.
Lecomte, J. Phys. A 41, 505001 (2008)] can be reinterpreted via the two
corresponding quantum Hamiltonians that are supersymmetric partners of each
other, with the same energy spectra. We describe the consequences for the
spectral decomposition of the boundary-driven dynamics, and we give explicit
expressions for the Kemeny times needed to converge towards the non-equilibrium
steady states. Finally, we analyze the large deviations at various levels for
empirical time-averaged observables over a large time-window $T$. We start with
the always explicit Level 2.5 concerning the joint distribution of the
empirical density and of the empirical flows, and we then consider the
contractions towards lower levels. In particular, the rate function for the
empirical current alone can be explicitly computed via the contraction from the
Level 2.5 using the properties of the associated quantum supersymmetric
Hamiltonians.
