In this paper, we established the Freidlin–Wentzell-type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.
"Large deviation principles for first-order scalar conservation laws with stochastic forcing." Ann. Appl. Probab. 30 (1) 324 - 367, February 2020. https://doi.org/10.1214/19-AAP1503