Probabilistic approximation and inviscid limits for 1-D fractional conservation laws

(x) when u(0, x) is the cumulative distribution function of a signed measure on IR. We associate a nonlinear martingale problem with the Fokker-Planck equation obtained by spatial differentiation of the conservation law. After checking uniqueness for both the conservation law and the martingale problem, we prove existence thanks to a propagation of chaos result for systems of interacting particles with fixed intensity of jumps related to ν. The empirical cumulative distribution functions of the particles converge to the solution of the conservation law. Finally, when the intensity of jumps vanishes (ν → 0) as the number of particles tends to +∞, we obtain that the empirical cumulative distribution functions converge to the unique entropy solution of the inviscid (ν = 0) conservation law.