  The measurement of higher order cumulants of the current noise generated by a
nonlinear mesoscopic conductor using a Josephson junction as on-chip detector
is investigated theoretically. The paper addresses the regime where the noise
of the mesoscopic conductor initiates activated escape of the Josephson
detector out of the zero-voltage state, which can be observed as a voltage
rise. It is shown that the deviations from Johnson-Nyquist noise can mostly be
accounted for by an effective temperature which depends on the second noise
cumulant of the conductor. The deviations from Gaussian statistics lead to
rather weak effects and essentially only the third cumulant can be measured
exploiting the dependence of the corrections to the rate of escape from the
zero-voltage state on the direction of the bias current. These corrections
vanish as the bias current approaches the critical current. The theory is based
on a description of irreversible processes and fluctuations in terms of state
variables and conjugate forces. This approach, going back to work by Onsager
and Machlup, is extended to account for non-Gaussian noise, and it is shown
that the thermodynamically conjugate force to the electric charge plays a role
similar to the counting field introduced in more recent approaches to describe
non-Gaussian noise statistics. The theory allows to obtain asymptotically exact
results for the rate of escape in the weak noise limit for all values of the
damping strength of the Josephson detector. Also the feedback of the detector
on the noise generating conductor is fully taken into account by treating both
coupled mesoscopic devices on an equal footing.
