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

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**Upcoming talks**** **

**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: Jiang 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.

**Past talks**

**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****3****. 1****5****:0****0****(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

## Abstract

随机串并联图由Hambly和Jordan于2004年引入。我将关于图上的距离做些简单讨论。报告基于同陈新兴、Bernard Derrida、Thomas Duquesne的合作工作。