We prove convergence results of the simulation of the density solution to the McKean-Vlasov equation, when the measure variable is in the drift. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel-type estimator. In particular, we give a generalised Bernstein inequality for Euler schemes with interacting particles and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters. This is a joint work with M. Hoffmann.

Standard symmetric α-stable cylindrical processes in Hilbert spaces are the natural generalisation of the analogue processes in Euclidean spaces. However, like standard Brownian motions, standard symmetric α-stable processes in finite dimensions can only be generalised to infinite dimensional Hilbert spaces as cylindrical processes, i.e. processes in a generalised sense (of Gel’fand and Vilenkin (1964) or Segal (1954))  not attaining values in the underlying Hilbert space.

In this talk, we briefly introduce the theory of stochastic integrals with respect to standard symmetric α-stable cylindrical processes. As these processes exist only in the generalised sense, introducing a stochastic integral requires an approach different to the classical one by semi-martingale decomposition. The main result presented in this talk is the existence of a solution to an abstract evolution equation driven by a standard symmetric α-stable cylindrical process. The main tool for establishing this result is a Yosida approximation and an Itô formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical α-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.