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.

We consider stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter less than 1/2. The drift is a measurable function of time and space which belongs to a certain Lebesgue space. Under subcritical regime, we show that a strong solution exists and is unique in path-by-path sense. When the noise is formally replaced by a Brownian motion, our results correspond to the strong uniqueness result of Krylov and Roeckner (2005). Our methods forgo standard approaches in Markovian settings and utilize Lyons' rough path theory in conjunction with recently developed tools. Joint work with Toyomu Matsuda and Oleg Butkovsky.

Until the 2010’s the only « comparison geometry » result for compact Riemannian manifolds (M^n,g) with scal≥n(n-1) was Greene’s upper bound on the injectivity radius. Moreover, it is known that classical metric invariants (volume, diameter) cannot be controlled by a lower bound on the scalar curvature alone. It has only recently been discovered that some more subtle invariants, such as 2-systoles, can be controlled under a lower bounds on scal provided M has enough topology. We will present some results of Bray-Brendle-Neves (in dim 3), Zhu (in dim≤7) for S^2xT^(n-2), some version for S^2xS^2 and some conjecture with more general topology which we show to hold true under the additional assumption of Kaehlerness.

In a classical work, Bowen and Margulis proved the equidistribution of
closed geodesics in any hyperbolic manifold. Together with Jeremy Kahn
and Vladimir Marković, we asked ourselves what happens in a
three-manifold if we replace curves by surfaces. The natural analog of a
closed geodesic is then a minimal surface, as totally geodesic surfaces
exist only very rarely. Nevertheless, it still makes sense (for various
reasons, in particular to ensure uniqueness of the minimal
representative) to restrict our attention to surfaces that are almost
totally geodesic.

The statistics of these surfaces then depend very strongly on how we
order them: by genus, or by area. If we focus on surfaces whose *area*
tends to infinity, we conjecture that they do indeed equidistribute; we
proved a partial result in this direction. If, however, we focus on
surfaces whose *genus* tends to infinity, the situation is completely
opposite: we proved that they then accumulate onto the totally geodesic
surfaces of the manifold (if there are any).

We propose a novel framework for exploring weak and $L_2$ generalization errors of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk minimization problem and establish generic conditions under which the generalization error convergence rate, when training on a sample of size $n$ , is $\matcal{O}(1/n)$. In the context of supervised learning with a one-hidden layer neural network in the mean-field regime, these conditions are reflected in suitable integrability and regularity assumptions on the loss and activation functions.