Donsker theorem for rough paths

Rough paths were introduced by Lyons in 1998 to cope with the non-continuity of the Itô map (the map which sends a sample-path of the Brownian motion to the solution of an SDE driven by this path). The so-called enriched Brownian motion is the couple made by the Brownian itself and its Lévy area. It is thus a natural question to enrich the Donsker theorem and see if the adjunction of this new component does change the convergence rate we established earlier.

Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry

The Poisson point process is the first brick on which the stochastic geometry is built upon. Since we have a very well developed Malliavin calculus for this process, it is rather straightforward to apply the Stein-Malliavin-Dirichlet to this setting. This gives raise to this beautiful paper. We investigate several multi-points transformations of point processes which lead to a Poissonian limit. Once again, a cornerstone of the calculations is the Stein representation formula of the Rubsintein distance as an integral along the path of an Ornstein-Ulhenbeck type process.

Stein-Dirichlet-Malliavin method

Stein-Dirichlet-Malliavin method

Stein's method for Brownian approximations