  In this paper we develop the Non-Equilibrium Green's Function (NEGF)
formalism for a dissipative molecular junction that consists of a central
molecular system with one-dimensional electronic transport coupled to a phonon
environment and attached to multiple electronic leads. Our approach is
partitionless - initial preparation of the system places the whole system in
the correct canonical equilibrium state - and is valid for an external bias
with arbitrary time dependence. Using path integrals as an intermediary tool,
we apply a two-time Hubbard-Stratonovich transformation to the phonon influence
functional with mixed real and imaginary times to obtain an exact expression
for the electronic density matrix at the expense of introducing coloured
Gaussian noises whose properties are rigorously derived from the environment
action. This results in a unique stochastic Hamiltonian on each branch of the
Konstantinov-Perel' contour (upper, lower, vertical) such that the time
evolution operators in the Liouville equation no longer form a Hermitian
conjugate pair, thus corresponding to non-Hermitian dynamics. To account for
this we develop a generalized three-time NEGF which is sensitive to all
branches of the contour, and relate it to the standard NEGF in the absence of
phonons via a perturbative expansion of the noises. This approach is exact and
fully general, describing the non-equilibrium driven dynamics from an initial
thermal state while subject to inelastic scattering, and can be applied to
non-Hermitian dynamics in general.
