  We propose to characterize L\'evy-distributed stochastic fluctuations through
the measurement of the average voltage drop across a current-biased Josephson
junction. We show that the noise induced switching process in the Josephson
washboard potential can be exploited to reveal and characterize L\'evy
fluctuations, also if embedded in a thermal noisy background. The measurement
of the average voltage drop as a function of the noise intensity allows to
infer the value of the stability index that characterizes L\'evy-distributed
fluctuations. An analytical estimate of the average velocity in the case of a
L\'evy-driven escape process from a metastable state well agrees with the
numerical calculation of the average voltage drop across the junction. The best
performances are reached at small bias currents and low temperatures,
\emph{i.e.}, when both thermally activated and quantum tunneling switching
processes can be neglected. The effects discussed in this work pave the way
toward an effective and reliable method to characterize L\'evy components
eventually present in an unknown noisy signal.
