  In this work we study by numerical methods the phase dynamics in ballistic
graphene-based short Josephson junctions. The supercurrent through a graphene
junction shows a non-sinusoidal phase-dependence, unlike a conventional
junction ruled by the well-known d.c. Josephson relation. A
superconductor-graphene-superconductor system exhibits superconductive quantum
metastable states similar to those present in normal current-biased JJs. We
explore the effects of thermal and correlated fluctuations on the escape time
from these metastable states, when the system is stimulated by an oscillating
bias current. As a first step, the analysis is carried out in the presence of
an external Gaussian white noise source, which mimics the random fluctuations
of the bias current. Varying the noise intensity, it is possible to analyze the
behavior of the escape time from a superconductive metastable state in
different temperature regimes. Noise induced phenomena, such as resonant
activation and noise induced stability, are observed. The study is extended to
the case of a coloured Gaussian noise source, analyzing how the escape time
from the metastable state is affected by correlated random fluctuations for
different values of the noise correlation time.
