# 2022 Probability Webinar

2022 Probability Webinar will be held via Zoom. We are looking forward to seeing you online!

## Venue

### Webinar by Zoom

If you would like to attend this webinar, please contact Bin XIE by e-mail: bxie AT shinshu-u.ac.jp in advance. After that, you will receive an email containing information of Zoom for joining the webinar.

## Program

### January 5, 2022 (Wednesday)

- 14:00 ～ 14:50 Yuzuru INAHAMA (Kyushu University)

Large deviations for small noise hypoelliptic diffusion bridges on sub-Riemannian manifolds**Abstract：**In this talk we discuss a large deviation principle of Freidlin-Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we combine rough path theory, manifold-valued Malliavin calculus, and quasi-sure analysis (which is a potential theoretic part of Malliavin calculus).

### February 3, 2022 (Thursday）

- 17:30 ～ 18:20 Ludovic Goudenège (CNRS FR-3487 & CentraleSupélec, Université Paris-Saclay)

Invariant measure for Stochastic alpha-Navier-Stokes equation with trace-class noise**Abstract：**Following the existence of a strong solution for the stochastic alpha-Navier-Stokes equation, we propose a numerical scheme based on the finite element method to approximate the solution in some domain of 2 or 3 dimensions. We prove the convergence of the numerical scheme when the time step and the spatial mesh size converge to 0. Moreover, in a particular case, we can obtain the convergence of the numerical scheme when alpha converges to 0 to recover the classical stochastic Navier-Stokes equation in 2D domains. This is joint work with Jad Doghman from Fédération de Mathématiques de CentraleSupélec, CNRS FR-3487, Université Paris-Saclay, France. - 18:30 ～ 19:20 Ludovic Goudenège (CNRS FR-3487 & CentraleSupélec, Université Paris-Saclay)

Numerical approximation of stochastic alpha-Navier-Stokes equation**Abstract：**Following the existence of a strong solution for the stochastic alpha-Navier-Stokes equation, we propose a numerical scheme based on the finite element method to approximate the solution in some domain of 2 or 3 dimensions. We prove the convergence of the numerical scheme when the time step and the spatial mesh size converge to 0. Moreover, in a particular case, we can obtain the convergence of the numerical scheme when alpha converges to 0 to recover the classical stochastic Navier-Stokes equation in 2D domains. This is joint work with Jad Doghman from Fédération de Mathématiques de CentraleSupélec, CNRS FR-3487, Université Paris-Saclay, France.

### February 10, 2022 (Thursday)

- 13：00～14：30 Hidetoshi NAKAGAWA (Hitotsubashi University）

TBA - 14：45～16：00 Hidetoshi NAKAGAWA (Hitotsubashi University）

TBA

### February 16, 2022 (Wednesday)

- 13:30 ～ 15:00 Takehiko MORITA (Osaka University )

TBA - 15:10 ～ 16:40 Takehiko MORITA (Osaka University )

TBA - 17:30 ～ 18:20 Ludovic Goudenège (CNRS FR-3487 & CentraleSupélec, Université Paris-Saclay)

Simulating intermittency by irregular noise in SDEs**Abstract：**When we aim at numerically describing the solution of Navier-Stokes, we need to add to the classical numerical schemes some expected physical behavior. In particular, if many particles are immerged in a fluid, they have to exhibit intermittency behavior in their statistical quantities of interest. The classical approach adds random noise in the deterministic path of the particles following the fluid in order to recover the intermittency. But classical Gaussian noises like Brownian motion are not enough irregular to describe all the statistical descriptors of intermittency. We propose a way to build a very singular noise that can recover the intermittency. Moreover, it is also highly efficient in numerical approaches. The idea consists in describing a large family of Gaussian noises in a unified framework that can have a singular limit. This framework supports the fractional Brownian motions with Hurst parameter in (0,1), but also the singular limit as H goes to 0. The next steps consist of mixing all these noises multiplicatively to create the well-known energy cascade in Kolmogorov's description of intermittency, whose expected mathematical object is the Gaussian Multiplicative Chaos. This is joint work with Alexandre Richard from Fédération de Mathématiques de CentraleSupélec, CNRS FR-3487, MICS, Université Paris-Saclay, France, and Roxane Letournel from EM2C, CNRS UPR-288, CentraleSupélec, Université Paris-Saclay, France. - 18:30 ～ 19:20 Martin Grothaus (Technische Universität Kaiserslautern)

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations**Abstract：**Motivated by problems from Industrial Mathematics we further developed the concepts of hypocoercivity. The original concepts needed Poincaré inequalities and were applied to equations in linear finite-dimensional spaces. Meanwhile we can treat equations in manifolds or even infinite dimensional spaces. The condition giving micro- and macroscopic coercivity we could relax from Poincaré to weak Poincaré inequalities. In this talk an overview and many examples are given.

## Supports

This event is partially supported by Japan Society for the Promotion of Science (JSPS Kakenhi), Grant-in-Aid for Scientific Research (C) 20K03627 (PI：Bin XIE).