  In this paper, we consider the Cauchy problem for the nonlinear fractional
conservation laws driven by a multiplicative noise. In particular, we are
concerned with the well-posedness theory and the study of the long-time
behavior of solutions for such equations. We show the existence of desired
kinetic solution by using the vanishing viscosity method. In fact, we establish
strong convergence of the approximate viscous solutions to a kinetic solution.
Moreover, under a nonlinearity-diffusivity condition, we prove the existence of
an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally,
we show the uniqueness and ergodicity of the invariant measure.
