It is shown that a one-channel coherent conductor in an Ohmic environment can be mapped to the impurity problem in a Tomonaga-Luttinger liquid. This allows one to determine nonperturbatively 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.
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