  We present a theoretical analysis of the effect of inelastic electron
scattering on current and its fluctuations in a mesoscopic quantum dot (QD)
connected to two leads, based on a recently developed nonperturbative technique
involving the approximate mapping of the many-body electron-phonon coupling
problem onto a multichannel single-electron scattering problem. In this, we
apply the B\"uttiker scattering theory of shot noise for a two-terminal
mesoscopic device to the multichannel case with differing weight factors and
examine zero-frequency shot noise for two special cases: (i) a single-molecule
QD and (ii) coupled semiconductor QDs. The nonequilibrium Green's function
method facilitates calculation of single-electron transmission and reflection
amplitudes for inelastic processes under nonequilibrium conditions in the
mapping model. For the single-molecule QD we find that, in the presence of the
electron-phonon interaction, both differential conductance and differential
shot noise display additional peaks as bias-voltage increases due to
phonon-assisted processes. In the case of coupled QDs, our nonperturbative
calculations account for the electron-phonon interaction on an equal footing
with couplings to the leads, as well as the coupling between the two dots. Our
results exhibit oscillations in both the current and shot noise as functions of
the energy difference between the two QDs, resulting from the spontaneous
emission of phonons in the nonlinear transport process. In the "zero-phonon"
resonant tunneling regime, the shot noise exhibits a double peak, while in the
"one-phonon" region, only a single peak appears.
