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We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolicequation in any space dimension with nonlocal, nonlinear, and possiblydegenerate diffusion terms. The equations are nonlocal because they involvefractional diffusion operators. We adapt the notion of stochastic entropysolution and provide a new technical framework to prove the uniqueness. Theexistence proof relies on the vanishing viscosity method. Moreover, usingbounded variation (BV) estimates for vanishing viscosity approximations, wederive an explicit continuous dependence estimate on the nonlinearities andderive error estimate for the stochastic vanishing viscosity method. Inaddition, we develop uniqueness method "a la Kruzkov" for more generalequations where the noise coefficient may depends explicitly on the spatialvariable.

A fractional degenerate parabolic-hyperbolic Cauchy problem with noise