Stochastic fractional conservation laws: large deviation principle, central limit theorem and moderate deviation principle

In this article, we establish the Freidlin-Wentzell type large deviation principle for stochastic fractional conservation laws with small multiplicative noise in the kinetic formulation framework. The weak convergence method and the doubling of variables method play a crucial role. As a consequence, we also establish the central limit theorem and moderate deviation principle for the underlying problem under constant initial data.