  We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic
equation in any space dimension with nonlocal, nonlinear, and possibly
degenerate diffusion terms. The equations are nonlocal because they involve
fractional diffusion operators. We adapt the notion of stochastic entropy
solution and provide a new technical framework to prove the uniqueness. The
existence proof relies on the vanishing viscosity method. Moreover, using
bounded variation (BV) estimates for vanishing viscosity approximations, we
derive an explicit continuous dependence estimate on the nonlinearities and
derive error estimate for the stochastic vanishing viscosity method. In
addition, we develop uniqueness method "a la Kruzkov" for more general
equations where the noise coefficient may depends explicitly on the spatial
variable.
