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It is shown that a one-channel mesoscopic conductor in an ohmic environmentcan be mapped to the problem of a backscattering impurity in aTomonaga-Luttinger liquid (TLL). This allows to determine non perturbativelythe effect of the environment on $I-V$ curves, and to find an exactrelationship between dynamic Coulomb blockade and shot noise. We investigatecritically how this relationship compares to recent proposals in theliterature. The full counting statistics is determined at zero temperature.Comment: 5 pages, 2 figures, shortened version for publication in Phys. Rev.  Let

One-Channel Conductor in an Ohmic Environment: Mapping to a Tomonaga-Luttinger Liquid and Full Counting Statistics