We propose to characterize Levy-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 Levy 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 Levy-distributed fluctuations. An analytical estimate of the average velocity in the case of a Levy-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 Levy components eventually present in an unknown noisy signal.
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