  We are concerned with multidimensional stochastic balance laws. We identify a
class of nonlinear balance laws for which uniform spatial $BV$ bounds for
vanishing viscosity approximations can be achieved. Moreover, we establish
temporal equicontinuity in $L^1$ of the approximations, uniformly in the
viscosity coefficient. Using these estimates, we supply a multidimensional
existence theory of stochastic entropy solutions. In addition, we establish an
error estimate for the stochastic viscosity method, as well as an explicit
estimate for the continuous dependence of stochastic entropy solutions on the
flux and random source functions. Various further generalizations of the
results are discussed.
