Contact: Wei Liu ( wliu.math@whu.edu.cn), Chaoen Zhang (chaoenzhang@hit.edu.cn)

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**Upcoming talks**

**June 30, ****202****3****. 14****:3****0****(Beijing time) **

Speaker: Jinqiao Duan（Great Bay University）

Title: A Joyful Journey in Random Dynamical Systems

## Abstract

Complex dynamical systems are often under random fluctuations. The noisy fluctuations may be Gaussian or non-Gaussian, which are usually modeled by Brownian motion or Levy motion, respectively. Stochastic differential equationsare appropriate mathematical models for these random dynamical systems.

The speaker will present an overview about recent advances in random dynamical systems,including random interacting particle dynamics, stochastic Hamiltonian dynamics, and probabilistic approaches and data science method.for investigating transition phenomena.

**June 30, ****202****3****. 15****:45****(Beijing time) **

Speaker: Christian Leonard (Universite Paris Nanterre)

Title: Time reversal and the dynamics of diffusion processes

## Abstract

During the sixties, Ed Nelson initiated the mathematical investigation of the dynamical properties of the Brownian motion. Two key ingredients of his approach were time reversal and stochastic derivatives, allowing the introduction of the notions of current and osmotic velocities.

ln the same spirit, we present a generalization and an extension of his results under some finite entropy assumption. In particular, the entropic interpolations of a diffusion process between two prescribed marginals are shown to solve some least action principle in the Otto-Wasserstein space (equipped with the Riemannian metric of the quadratic optimal transport). Some well-known consequences in terms of Otto-Wasserstein gradient flows and contraction inequalities are recovered.

Some of these results were obtained in collaboration with P. Cattiaux, G Conforti and I. Gentil.

**Past talks**

**June 25,**** ****202****3****. 15****:3****0****(Beijing time) **

Speaker: Qi Lv（Sichuan University）

Title: Recent Progresses on Optimal Control of Stochastic Distributed Parameter Systems

## Abstract

In this talk, I present some recent progresses on optimal control of stochastic distributed parameter systems, including Pontryagin-type maximum principle, dynamic programming and linear quadratic optimal control problem.

**June 16,**** ****202****3****. 15****:3****0****(Beijing time) **

Speaker: Louis Sharrock （Lancaster University，UK）

Title:Parameter Estimation for the McKean-Vlasov Stochastic Differential Equation

## Abstract

McKean-Vlasov SDEs arise in many applications, including mathematical biology, social sciences, and machine learning. In this talk, wewill discuss parameter estimation for a cKean-Vlasov SDE and theassociated system of weakly interacting particles. We first analyse theasymptotic properties of the offline maximum likelihood estimator. Wethen propose a new online estimator, which evolves according to acontinuous-time stochastic gradient descent algorithm on the asymptoticog-likelihood of the interacting particle system. We obtain variousconvergence results for this estimator, under assumptions which guaranteeergodicity and uniform-in-time propagation chaos. Our theoretical resultsare supported via several numerical examples, including a toy linear meanfield model a stochastic Kuramoto model. and a stochastic opiniondynamics model.

**June 9,**** ****202****3****. 15****:3****0****(Beijing time) **

Speaker: Chenlin Gu（Tsinghua University）

Title:An iterative algorithm for Dirichlet problem withrandomconductance

## Abstract

The Dirichlet problem with oscillating coefficients requires a lot ofcalculation resources in general. We present a new algorithm invented by SArmstrong, A. Hannukainen, T. Kuusi, J-C. Mourrat to solve this problemquickly in the context of random conductance. This result makes use of thequantitative homogenization theory and the algorithm can also be generalized tothe percolation cluster setting.

**June 2,**** ****202****3****. 14****:3****0****(Beijing time) **

Speaker: Jian Ding（Peking University）

Title:Repeatedemergence of 4/3-exponent

## Abstract

In this talk, I will describe the emergence of the 4/3-exponent in two seem-ingly unrelated models: random distance of Liouville quantum gravity and correlation length for the two-dimensional random field Ising model. I will then explain that such 4/3-exponent, while being unexpected among respective communities even from a physics perspective, has in fact been hinted in Leighton-Shor(1989) and Talagrand (2014) where the 4/3-exponent emerges in a random matching problem Finally, I will present the heuristic computation which leads tothe emergence of the 4/3-exponent. Based on a joint work with Subhajit Goswami and a joint work with Mateo Wirth.

**June 2,**** ****202****3****. 15****:45****(Beijing time) **

Speaker: Jian Song（Shandong University）

Title:Scaling limit of a long-range random walk in time-correlated random environment

## Abstract

This paper concerns a long-range random walk in random environment indimension 1 + 1. where the environmental disorder is independent in space butcorrelated in time. We prove that the rescaled partition function converges weakly to the Stratonovich solution of a fractional stochastic heat equation with multi.plicative Gaussian noise which is white in space and colored in time. This is aoint work with Guanglin Rang and Meng Wang.

**May 26,**** ****202****3****. 15****:3****0****(Beijing time) **

Speaker: Peng Chen（Nanjing University of Aeronautics and Astronautics）

Title:A Refined Lindeberg Principle and Its Applications

## Abstract

We view the classical Lindeberg principle in a homogeneous Markov proc-ess setting to establish a probability approximation framework by the associatedIto's formula and Markov operator. As applications, we study the error bounds ofthe following three approximations: multivariate normal approximation, approx1.mating a family of online stochastic gradient descents (SGDs) by a stochasticdifferential equation (SDE) driven by multiplicative Brownian motion. andEuler-Maruyama (EM) discretization for SDEs driven by alpha-stable process.Furthermore, we extend the above approximation framework to the inhomo-geneous case and consider the variable-step E-M approximations of regime-switching jump diffusion processes. This talk is based on the joint works with OM. Shao. Z.G.Su C. Deng and L.Xu .

**May 19,**** ****202****3****. 15****:3****0****(Beijing time) **

Speaker: Yuling Jiao（Wuhan University）

Title: Theoretical study on deep learning: approximation, generalization and generation

## Abstract

In the first part of this talk, I will discuss some theoretical studies on deeplearning with a focus on approximation, generalization. In particular, I willcover error analysis with over-parameterization. In the second part, I will delveinto the concept of Gaussian stochastic interpolations and their applications,including the derivation of functional inequality with dimension-free constantsas well as sampling and generative learning.

**May 12,**** ****202****3****. 16****:0****0****(Beijing time) **

Speaker: Lingyan Cheng（Nanjing University of Science and Technology）

Title: A new algorithm for approximation of stochastic differential equations viajump processes

## Abstract

**May 5,**** ****202****3****. 15****:3****0****(Beijing time) **

Speaker: Yaozhong Hu（University of Alberta）

Title: Mean-Field Super-Brownian Motions

## Abstract

## The mean field stochastic partial differential equation (SPDEcorresponding to a mean field super-Brownian motion (sBm) is obtainedand studied. In this mean field sBm, the branching particle lifetime isallowed to depend upon the probability distribution of the sBm itselfproducing an SPDE whose space-time white noise coefficient has, inaddition to the typical sBm square root, an extra factor that is a functionof the probability law of the density of the mean field sBm. This novelmean field SPDE is thus motivated by population models where thingslike overcrowding and isolation can affect growth. A two step approximation method is employed to show the existence for this SPDE undergeneral conditions. Then, mild moment conditions are imposed to getuniqueness. Finally, smoothness of the SPDE solution is established undera further simplifying condition.

**April 28, 202****3****. 14****:0****0****(Beijing time) **

Speaker: Deng Zhang（Shanghai Jiao Tong University)

Title: Multi solitary waves to stochasticnonlinear Schrodinger equations

## Abstract

In this talk we will present the recent work on the multisolitary waves to stochastic nonlinear Schrodinger equations drivenby linear multiplicative noise, in both the mass critical andsubcritical cases. Unlike in the deterministic case, the existenceof stochastic multi-solitons cannot be obtained from that ofstochastic multi-bubble blow-up solutions, due to the absence ofpseudo-conformal invariance. We present a constructive proof byutilizing the rescaling approach and the modulation method. Theconstructed multi-solitons behave asymptotically as a sum offinitely many solitary waves, and the convergence rate of theremainders can be of either exponential or polynomial type, whichreflects the effects of noise on the asymptotical behavior ofsolutions.

**April 21,**** ****202****3****. 1****5****:3****0****(Beijing time) **

Speaker: Weijun Xu（Peking University)

Title: 路径的signature变换及其反问题

## Abstract

## 空间中路径的 signature 变换由该路径的迭代积分给出，取值于张量代数。其始于上世纪 50 年代陈国才的一系列工作。周炜良早年的一个定理在其中也有重要推论。 近年来，signature 也系统地出现在数据分析中，主要被用于识别曲线类数据的特征。关于 signature，我们可以问三个基本的问题：1. signature 变换的像是什么？即张量代数中什么样的元素可以经由路径的 signature 变换得到。

2. signature 变换是否是单射？即给定某个 signature，其对应的原路径是否唯一。

3. 反问题：如果唯一，那么给定 signature，如何（近似地）找出其对应的原路径。

关于第一个问题，有一些很自然的必要条件，但它们的充分性未知。对于后两个问题，近年来有了不错的进展。我将汇报一下这些进展，以及下一步期望解决的问题。

**April 7, April 14, 202**

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**(Beijing time)**

Speaker: Zhongmin Qian（Oxford University)

Title: Stochastic methods in computational fluiddynamics and turbulence

## Abstract

In these talks I am going to present a small part of the science suggested inthe title, and I aim to cover some ideas of applying stochastic calculus to Computational Fluid Dynamics (CFD) which are developed recently by various authors including some of my coauthors and myself: Prof. Endre Süli (Oxford), Prof. Youchun Qiu (Toulouse, France), Profianglun Wu (Swansea), Prof. Mingyu Xu (Fudan), Dr Jiawei Li (Edinburgh), Mr YihuangZhang (Oxford Dphil student), based on the following papers:

1. Tracking the vortex motion by using Brownian fluid particles, Phys. Fluids 33, 105113 (2021).

2. McKean-Vlasov type stochastic differential equations arising from the random vortex method Partial Differential Equations and Applications (2022) 3:7

3. Random vortex dynamics via functional stochastic differential equations. Proc. R. Soc. A 478: 70270030.

4. https://arxi.org/pdf/2206.05198.pdf

5.https://arxiw.org pdf/230317260.pdf

Lecture 1: In this talk I will explain the idea how to formulate (income-pressible) fluid dynamic equations into McKean-Vlasov type mean filed equations, and outline some Mathematical questions which may be worthy to explore. As an example, I will present a new methodfor solving a related McKean-Vlasov SDEs by means of weak solution method.

Lecture 2: In this talk I will first establish the duality of conditional laws among diffusionprocesses associated with solenoidal vector fields, and establish new functional integral representtation theorem for a class of linear (but temporal non-homogeneous) parabolic equationsFinally we apply the new representation theorem for the study of incompressible fluid flowspast a solid wall.

**March 31, 202****3****. 1****5****:****3****0****(Beijing time) **

Speaker: Xin Chen（Shanghai Jiao Tong University)

Title: Heat kernel estimates for Schodinger operators

## Abstract

We will introduce some probabilistic methods to study heatkernel estimates for Schodinger operators L--A+V, based onwhich the interaction between the behaviors of Brownian motionsBt}tzo and the potential V will be applied. Our results include thecase that V is unbounded or V is decaying to 0 at infinity.Moreover, two-sided Green's function estimates associated withL--A+V are also obtained.

**March 24, 202****3****. 1****5****:****3****0****(Beijing time) **

Speaker: Dejun Luo（Academy of Methematics and System Science Chinese Academy of Sciences)

Title: Some recent results on regularization by transport noise for fluid dynamical models

## Abstract

According to heuristic arguments and some recent rigorous studies, transportnoise is a physically well motivated random perturbation to fluid dynamicalequations. In this talk we introduce several results on regularizing effects of transportnoise, including suppression of blow-up for the vorticity form of 3D Navier-Stokesequations and asymptotic weak uniqueness for stochastic 2D Euler equation. Wealso mention regularizing effects of transport-type noise to other models such as theinviscid Leray-alpha model and the dyadic model for turbulence.

**March 17, 202****3****. 1****5****:****3****0****(Beijing time) **

Speaker: Zhan Shi（Academy of Methematics and System Science Chinese Academy of Sciences)

Title: Connection probabilities for 2D critical lattice model

## Abstract

Conformal invariance of critical lattice models in two-dimensional has been vigorously studied for decades. The first example where the conformal invariance was rigorously verified was the planar uniform spanning tree (together with loop-erased random walk), proved by Lawler, Schramm and Werner around 2000. Later, the conformal invariance was also verified for Bernoulli percolation (Smirnov 2001), level lines of Gaussian free field (Schramm-Sheffield 2009), and Ising model and FK-Ising model (Chelkak-Smirnov et al 2012). In this talk, we focus on connection probabilities of these critical lattice models in polygons with alternating boundary conditions.

This talk has two parts.

• In the first part, we consider critical Ising model and give the crossing probabilities of multiple interfaces. Such probabilities are related to solutions to BPZ equations in conformal field theory.

• In the second part, we consider critical random-cluster model with cluster weight q\in (0,4) and give conjectural formulas for connection probabilities of multiple interfaces. The conjectural formulas are proved for q=2, i.e. the FK-Ising model.

**March 10, 202****3****. 1****5****:****3****0****(Beijing time) **

Speaker: Xiangdong Li（Academy of Methematics and System Science Chinese Academy of Sciences)

Title: On W-entropy formulas for Langevin deformations over Wasserstein space

## Abstract

In this talk, I will briefly review the history of optimal transport problem and Otto's Riemannian structure on the Wasserstein space of probability measures over a Riemannian manifold. Then I will introduce the Langevin deformations which interpolate the heat equation or the porous medium equation and the geodesic flow on the Wasserstein space. The W-entropy formulas are proved for the Langevin flows. Based on joint works with S. Li, R. Lei and Y.Z. Wang.

**March 3, 202****3****. 1****5****:****3****0****(Beijing time) **

Speaker: Zhan Shi（Academy of Methematics and System Science Chinese Academy of Sciences)

Title: 随机串并联图上的距离