  We study the transport properties of generic out-of-equilibrium quantum
systems connected to fermionic reservoirs. We develop a new method, based on an
expansion of the current in terms of the inverse system size and out of
equilibrium formulations such as the Keldysh technique and the Meir-Wingreen
formula. Our method allows a simple and compact derivation of the current for a
large class of systems showing diffusive/ohmic behavior. In addition, we obtain
exact solutions for a large class of quantum stochastic Hamiltonians (QSHs)
with time and space dependent noise, using a self consistent Born diagrammatic
method in the Keldysh representation. We show that these QSHs exhibit diffusive
regimes which are encoded in the Keldysh component of the single particle
Green's function. The exact solution for these QSHs models confirms the
validity of our system size expansion ansatz, and its efficiency in capturing
the transport properties. We consider in particular three fermionic models: i)
a model with local dephasing ii) the quantum simple symmetric exclusion process
model iii) a model with long-range stochastic hopping. For i) and ii) we
compute the full temperature and dephasing dependence of the conductance of the
system, both for two- and four-points measurements. Our solution gives access
to the regime of finite temperature of the reservoirs which could not be
obtained by previous approaches. For iii), we unveil a novel
ballistic-to-diffusive transition governed by the range and the nature (quantum
or classical) of the hopping. As a by-product, our approach equally describes
the mean behavior of quantum systems under continuous measurement.
