## Section: New Results

### Stochastic Analysis and probabilistic numerical methods

#### Particles approximation of mean-field SDEs

O. Bencheikh and Benjamin Jourdain analysed the rate of convergence of a system of $N$ interacting particles with mean-field rank based interaction in the drift coefficient and constant diffusion coefficient [16], [30]. They first adapted arguments by Kolli and Shkolnikhov to check trajectorial propagation of chaos with optimal rate ${N}^{-1/2}$ to the associated stochastic differential equations nonlinear in the sense of McKean. They next relaxed the assumption needed by Bossy to check convergence in ${L}^{1}\left(\mathbf{R}\right)$ of the empirical cumulative distribution function of the Euler discretization with step $h$ of the particle system to the solution of a one dimensional viscous scalar conservation law with rate $\mathcal{O}\left(\frac{1}{\sqrt{N}}+h\right)$. Last, they proved that the bias of this stochastic particle method behaves in $\mathcal{O}\left(\frac{1}{N}+h\right)$, which is confirmed by numerical experiments.

#### Abstract Malliavin calculus and convergence in total variation

In collaboration with L. Caramellino (University Tor Vergata) and with G. Poly (University of Rennes), V. Bally has settled a Malliavin type calculus for a general class of random variables, which are not supposed to be Gaussian (as it is the case in the standard Malliavin calculus). This is an alternative to the $\Gamma $ calculus settled by Bakry, Gentile and Ledoux. The main application is the estimate in total variation distance of the error in general convergence theorems. This is done in [29].

#### Invariance principles

As an application of the methodology mentioned above, V. Bally and coauthors have studied several limit theorems of Central Limit type - (see [14] and [15]. In particular they have estimate the total variation distance between random polynomials on one hand, and proved an universality principle for the variance of the number of roots of trigonometric polynomials with random coefficients, on the other hand.

#### Regularity of the low of the solution of jump type equations

V. Bally, L. Caramellino and G. Poly obtained some new regularity results for the solution of the 2 dimensional Bolzmann equation (see [13]). Moreover, in collaboration with L. Caramellino and A. Kohatsu Higa, V. Bally has started a research program on the regularity of the solutions of jump type equations. A first result in this sense is contained in [28].

#### Approximation of ARCH models

Benjamin Jourdain and Gilles Pagès (LPSM) are interested in proposing approximations of a sequence of probability measures in the convex order by finitely supported probability measures still in the convex order [35]. They propose to alternate transitions according to a martingale Markov kernel mapping a probability measure in the sequence to the next and dual quantization steps. In the case of ARCH models and in particular of the Euler scheme of a driftless Brownian diffusion, the noise has to be truncated to enable the dual quantization step. They analyze the error between the original ARCH model and its approximation with truncated noise and exhibit conditions under which the latter is dominated by the former in the convex order at the level of sample-paths. Last, they analyse the error of the scheme combining the dual quantization steps with truncation of the noise according to primal quantization.

#### Convergence of metadynamics

By drawing a parallel between metadynamics and self interacting models for polymers, B. Jourdain, T. Leliëvre (Cermics / ENPC) and P.-A. Zitt (LAMA) study the longtime convergence of the original metadynamics algorithm in the adiabatic setting, namely when the dynamics along the collective variables decouples from the dynamics along the other degrees of freedom. They also discuss the bias which is introduced when the adiabatic assumption does not holds [34].

#### Optimal transport

With V. Ehrlacher, D. Lombardi and R. Coyaud, Aurelien Alfonsi isworking on numerical a approximations of the optimal transport between two (or more) probability measures [26].

#### Generic approximation schemes for Markov semigroups.

A. Alfonsi and V. Bally have produced a general approximation scheme for Markov semigroups, based on random grids. This is a new approach to approximation schemes which is an alternative to the multi level method and the Romberg method [24].

#### Approximation with rough paths

A. Alfonsi and A. Kebaier are working on the approximation of some processes with rough paths.