Diffusion processes associated with nonlinear evolution equations for signed measures
The author is interested in probabilistic particle approximations of the solution of a nonlinear parabolic evolution equation starting at a bounded signed measure. His approach generalizes results obtained when the initial condition is a probability measure. One introduces a nonlinear martingale problem, under the solution \(P\) of which one recovers the solution of the partial differential equation. Here, the originality consists in giving to each sample path a signed weight which depends on the initial position. After dealing with the classical McKean-Vlasov equation, the author is interested in a viscous scalar conservation law. He proves uniqueness for the associated nonlinear martingale problem and obtains existence thanks to a propagation of chaos result for a system of weakly interacting diffusion processes. Then he studies the associated fluctuations and presents numerical results which confirm the theoretical rate of convergence.
scientific article; zbMATH DE number 1501446