  This paper presents a fractional generalized Cauchy process (FGCP) with an
additive and a multiplicative Gaussian white noise for describing subordinated
anomalous fluctuations. The FGCP displays intermittent dynamics during random
time durations, whose analytical representation is given by the It$\hat{\rm o}$
stochastic integral. The associated probability density function is given by
the Tsallis $q$-Gaussian distribution at the stationary state. The method of
fractional Feynman-Kac formula shows that weak ergodicity breaking of the FGCP
depends on the existence of the subordinator and/or the divergence of variance.
