  Continuity equations associated to continuous-time Markov processes can be
considered as Euclidean Schr\"odinger equations, where the non-hermitian
quantum Hamiltonian $\bold{H}={\bold{div}}{\bold J}$ is naturally factorized
into the product of the divergence operator ${\bold {div}}$ and the current
operator ${\bold J}$. For non-equilibrium Markov jump processes in a space of
$N$ configurations with $M$ links and $C=M-(N-1)\geq 1$ independent cycles,
this factorization of the $N \times N$ Hamiltonian ${\bold H}={\bold
I}^{\dagger}{\bold J}$ involves the incidence matrix ${\bold I}$ and the
current matrix ${\bold J}$ of size $M \times N$, so that the supersymmetric
partner ${\hat{\bold H}}= {\bold J}{\bold I}^{\dagger}$ governing the dynamics
of the currents living on the $M$ links is of size $M \times M$. To better
understand the relations between the spectral decompositions of these two
Hamiltonians $\bold{H}={\bold I}^{\dagger}{\bold J}$ and ${\hat {\bold H}}
={\bold J}{\bold I}^{\dagger}$ with respect to their bi-orthogonal basis of
right and left eigenvectors that characterize the relaxation dynamics towards
the steady state and the steady currents, it is useful to analyze the
properties of the Singular Value Decompositions of the two rectangular matrices
${\bold I}$ and ${\bold J} $ of size $M \times N$ and the interpretations in
terms of discrete Helmholtz decompositions. This general framework concerning
Markov jump processes can be adapted to non-equilibrium diffusion processes
governed by Fokker-Planck equations in dimension $d$, where the number $N$ of
configurations, the number $M$ of links and the number $C=M-(N-1)$ of
independent cycles become infinite, while the two matrices ${\bold I}$ and
${\bold J}$ become first-order differential operators acting on scalar
functions to produce vector fields.
