  It is shown that a one-channel mesoscopic conductor in an ohmic environment
can be mapped to the problem of a backscattering impurity in a
Tomonaga-Luttinger liquid (TLL). This allows to determine non perturbatively
the effect of the environment on $I-V$ curves, and to find an exact
relationship between dynamic Coulomb blockade and shot noise. We investigate
critically how this relationship compares to recent proposals in the
literature. The full counting statistics is determined at zero temperature.
