  Motivated by several experimental activities to detect charge noise produced
by a mesoscopic conductor with a Josephson junction as on-chip detector, the
switching rate out of its zero-voltage state is studied. This process is
related to the fundamental problem of thermal escape in presence of
non-Gaussian fluctuations. In the relevant case of weak higher than second
order cumulants, an effective Fokker-Planck equation is derived, which is then
used to obtain an explicit expression for the escape rate. Specific results for
the rate asymmetry due to the third moment of current noise allow to analyse
experimental data and to optimize detection circuits.
