Convergence of a stochastic particle approximation for fractional scalar conservation laws

In this paper, we are interested in approximating the solution to scalar conservation laws using systems of interacting stochastic particles. The scalar conservation law may involve a fractional Laplacian term of order α ∈ (0, 2]. When α ≤ 1 as well as in the absence of this term (inviscid case), its solution is characterized by entropic inequalities. The probabilistic interpretation of the scalar conservation is based on a stochastic differential equation driven by an α-stable process and involving a drift nonlinear in the sense of McKean. The particle system is constructed by discretizing this equation in time by the Euler scheme and replacing the nonlinearity by interaction. Each particle carries a signed weight depending on its initial position. At each discretization time we kill the couples of particles with opposite weights and positions closer than a threshold since the contribution of the crossings of such particles has the wrong sign in the derivation of the entropic inequalities. We prove convergence of the particle approximation to the solution of the conservation law as the number of particles tends to ∞ whereas the discretization step, the killing threshold and, in the inviscid case, the coefficient multiplying the stable increments tend to 0 in some precise asymptotics depending on whether α is larger than the critical level 1.