课题：国家重点研发项目“无穷维随机微分几何与Malliavin 分析理论”

编号：2020YFA0712702

组织者：中国科学院数学与系统科学研究院李向东

报告摘要

2025年10月24日

1、报告人：王凤雨 天津大学

报告题目：Stochastic intrinsic gradient flows on the Wasserstein space

摘要： We construct stochastic gradient flows on the Wasserstein space for a class of energy functionals,which includes the entropy functional and the Lyapunov function of generalized porous media equations. First,we define a class of Gaussian-based measures and weighted O-U Processes on the Wasserstein space,then use Dirichlet form techniques to construct quasi-regular Dirichlet forms associated with Gibbs measures induced by the energy functionals and Gaussian measures,and finally prove that the associated diffusion processes provide weak solutions to the stochastic intrinsic gradient flows. This is a joint work with Panpan Ren,Michael Rockner and Simon Wittmann.

2、报告人：吴付科 华中科技大学

报告题目：Systems of Singularly Perturbed Forward-Backward Stochastic Differential Equations and Control Problems

摘要：This paper focuses on systems of singularly perturbed forward-backward stochastic differential equations (FBSDEs) and control problems. Assuming Lipschitz continuity on the coefficients and allowing degeneracy in the diffusion terms, the solution of a two-time-scale FBSDE is shown to converge to the solution of an averaged FBSDE as a small parameter $\e$ tends to zero. Furthermore, it is shown that the value function of the singularly perturbed systems converges to the solution of a nonlinear partial differential equation (PDE). Furthermore,under additional conditions,it is demonstrated that the solution of the limit PDE is in fact the limit value function. These results provide insights into the convergence rate and extend existing results on the averaging principles for such stochastic control problems.

3、报告人：巫静 中山大学

报告题目：Limit theorems of SDEs and SFDEs with reflections

摘要：This talk is concerned with some problems of SDEs and SFDEs with reflections in non-smooth domains,including the wellposedness results,limit theorems and the support characterization.

4、报告人：韩邦先 山东大学

报告题目：Wasserstein重心问题的最新进展

摘要：在我与中国科大刘登宇和朱卓楠合作的第一篇关于Wasserstein重心的论文中，我们研究了存在性、正则性等问题，并提出了我们称之为“重心曲率维数条件”的概念。在这个报告中，我讲介绍我们的一些最新进展，包括Wasserstein重心版本的连续性方程和Benamou-Brenier公式，测度收缩性质（measure contraction property)等

5、报告人：张朝恩 哈尔滨工业大学

报告题目：The logarithmic Sobolev inequality and its nonlinear analogues

摘要：The entropy producing phenomenon is fundamental in statistical physics. It demonstrates the tendency of non-equilibrium states to equilibrium states as time evolves. In the mathematical study of kinetic theory, it is one of the key goals to describe this phenomenon quantitatively and rigorously. Logarithmic Sobolev inequalities and their nonlinear analogues have played an important role in this direction. In this talk, I will present results and questions about this functional inequality approach. I will start by introducing basic facts and some related questions/progresses about logarithmic Sobolev inequality. Then I will review results and questions about its nonlinear analogues. More precisely,I will focus on the following models: the McKean-Vlasov equation,the Landau equation and the Kac model.

6、报告人：王才士 西北师范大学

报告题目：The Canonical Process and Feynman-Kac Formula on the Infinite dimensional Hypercube

摘要：The infinite dimensional hypercube (IDH) is a novel example of locally infinite graphs and has found application in understanding Anderson localization and other related physical phenomena. In this talk,we would like to present some results on the IDH from a viewpoint of stochastic analysis. We first show that the canonical process on the IDH is a continuous-time Markov process with certain regularity. And then,with the canonical process as the main tool,we prove a Feynman-Kac formula on the IDH.

7、报告人：尹智 中南大学

报告题目：Outlier eigenvalues for full rank deformed single ring random matrices

摘要：We consider the full rank deformed single ring model. We investigate their eigenvalues (called outliers) that are outside the support of their limiting empirical spectral measure. This is a joint work with Ching-Wei Ho and Ping Zhong.

8、报告人：钱斌 常熟理工大学

报告题目：Gradient bounds for some hypoelliptic diffusions

摘要：Consider the SDE in $\mathbb{R}^n$ $ dX_t=(A(t)X_t+b(t))dt+\sigma(t)dB_t$where $B$ is a $d-$dimensional Brownian motion with $d\le n$ and $\sigma(t),A(t),b(t)$ are $L^{\infty}_{loc}(\mathbb{R})$ functions with values respectively in the matrix spaces of dimension $n\times d,n\times n,n\times 1$. Assume the covariance of $X_t$ is positive definite,we obtain the (right and reverse) Bakry-\'Emery inequality by coupling. Moreover,we can obtain Bismut formula by Mallivan calculus and coupling. Consequently,we can give Poincar\'e inequality,Log-Sobolev inequality,Wang-Harnack inequality and Transportation inequalities for the associated semigroup $P_t$.

9、报告人：胡泽春 四川大学

报告题目：Some new results of one-dimensional simple random walk

摘要：In this talk,we will introduce our recent results on favorite edges and rarely visited edges of one-dimensional simple symmetric random walk,and on favorite sites of one-dimensional asymmetric random walk based on the following three papers:

[1] C.-X. Hao,Z.-C. Hu,T. Ma,R. Song: Three favorite edges occurs infinitely often for one-dimensional simple random walk,Communications in Mathematics and Statistics,2025+,DOI: 10.1007/s40304-023-00382-2..

[2] Z.-C. Hu,X. Peng,R. Song,Y. Tan: The asymptotic behavior of rarely visited edges of the simple random walk,arXiv: 2310.16657v1 (2023).

[3] Z.-C. Hu,R. Song,G.-S. Zhou: Favorite sites of one-dimensional asymmetric simple random walk,In preparation.

10、报告人：李骥 华中科技大学

报告题目：orbital stability of breathers in the modified Camassa-Holm equation

摘要：The modified Camassa-Holm equation (mCH) with a cubic nonlinearity is an integrabel and nonlocal mathematical model for the unidirectional propagation of shallow- water waves. This study establishes the existence of time-periodic,spatially localized smooth-wave solutions,known as breathers,within a specific range of the linear dispersive parameter. By employing three rarely used conserved quantities,expressed in terms of the momentum variable m,it is demonstrated that breathers,as solutions to the mCH equation,are orbitally stable under perturbations in the Sobolev space H2.

2025年10月25日

1、报告人：  姚建峰 香港中文大学（深圳）数据科学学院

报告题目：  On eigenvalue distributions of large auto-covariance matrices

摘要： In this article,we establish a limiting distribution for eigenvalues of a class of autocovariance matrices. The same distribution has been found in the literature for a regularized version of these autocovariance matrices. The original nonregularized autocovariance matrices are noninvertible,thus introducing supplementary difficulties for the study of their eigenvalues through Girko’s Hermitization scheme. The key result in this paper is a new polynomial lower bound for a specific family of least singular values associated to a rank-defective quadratic function of a random matrix with independent and identically distributed entries. Another innovation from the paper is that the lag of the autocovariance matrices can grow to infinity with the matrix dimension.（joint work with 袁望钧，南方科技大学数学系）

2、报告人：吕琦 四川大学

报告题目： Some Applications of Malliavinian Calculus in Stochastic Control Theory

摘要： This talk is devoted to the fruitful interplay between Malliavin calculus and stochastic control theory. We first focus on a series of recent results in stochastic control,whose proofs rely crucially on techniques from Malliavin calculus. Subsequently,we present several open problems within Malliavin calculus itself that have arisen from these stochastic control applications.

3、报告人： 王振富 北京大学

报告题目:Kac’s program for the Landau equation

摘要： We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative N -particle system,obtained by passing to the grazing limit on Kac’s walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials,including the physically important Coulomb case. This provides the first resolution of propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions,and the first extension of Kac’s program to the Landau equation in the soft potential regime. The convergence is established in weak,Wasserstein,and entropic senses,together with strong L1 convergence. To handle the singularity of soft potentials,we extend the duality approach of Bresch-Duerinckx-Jabin and establish key functional inequalities,including an extended commutator estimate and a new second-order Fisher information estimate. Based on a joint work with Xuanrui Feng (PKU).

4、报告人： 吴波 复旦大学

报告题目：Functional inequalities for diffusions on path/loop spaces

摘要： In this talk,we first introduce recent developments of functional inequalities on the Riemannian path/loop space. After that,we will present associated functional inequalities with respect to diffusion Wiener measure and diffusion Bridge measure respectively

5、报告人： 胡军浩 中南民族大学

报告题目：Stabilization in distribution of periodic hybrid systems by discrete-time

state feedback control

摘要：Periodic hybrid stochastic differential equations (SDEs) have been widely used to model systems in many branches of science and industry which are subject to the following natural phenomena: (a) uncertainty and environmental noise,(b) abrupt changes in their structure and parameters, (c) periodicity. In many situations, it is inappropriate to study whether the solutions of periodic hybrid SDEs will converge to an equilibrium state (say, 0 or the trivial solution) but more appropriate to discuss whether the probability distributions of the solutions will converge to a stationary distribution, known as stability in distribution. Given a periodic hybrid SDE, which is not stable in distribution, can we design a periodic feedback control in the shift term based on state observations at discrete times so that the controlled SDE becomes stable in distribution? We will refer to this problem as stabilization in distribution by periodic feedback control. There is little known on this problem so far. This paper initiates the study in this direction.

6、报告人： 王冉 武汉大学

报告题目 Analysis of the gradient for the SPDEs with spatially-colored noise

摘要：For stochastic heat and wave equations driven by a centered Gaussian noise that is white in time and spatially homogeneous with covariance given by the Riesz kernel,we investigate the detailed behavior of approximated spatial and temporal gradients. As applications,we derive laws of the iterated logarithm and describe the asymptotic behavior of the q-variations of the solutions.

7、报告人： 成灵妍 南京理工大学

报告题目：Some results about the Multi-valued McKean-Vlasov SDEs with jumps

摘要：In this talk,we present several results of multivalued McKean-Vlasov stochastic differential equations (MMVSDEs) driven by Lévy noise under non-Lipschitz coefficients. Firstly,we rigorously establish the existence and uniqueness of strong solutions for this class of equations,overcoming challenges posed by the interplay between multivalued operators, measure-dependent coefficients, and discontinuous Lévy processes. Subsequently, we investigate the asymptotic behavior of small perturbations for the system. Utilizing the weak convergence approach, we derive Freidlin-Wentzell type large deviation principles (LDPs) and moderate deviation principles (MDPs) for MMVSDEs. Finally, we study the exponential ergodicity of these equations and establish convergence of corresponding invariant measure. This talk is based on the joint work with Caihong Gu,Wei Liu,and Fengwu Zhu.

8、报告人：黄乔 东南大学

报告题目： A tentative attempt to bridge Stochastic Geometric Mechanics & Stochastic Thermodynamics.

摘要:This talk systematically investigates the mathematical structure of path measures,both from a measure-theoretical perspective and through stochastic differential equations. The realization of path measures as Langevin systems hinges on the pivotal role of second-order Hamilton--Jacobi--Bellman equations,which form the foundation of stochastic geometric mechanics and applications in stochastic thermodynamics. We explore the emergence of the Onsager--Machlup functional in large deviation theory,the rates of entropy production in irreversible thermodynamic processes,and entropy minimization problems encoded in stochastic geometric mechanics. This talk is based on joint work with Dr. Jianyu Hu and Dr. Yuanfei Huang.

9、报告人：唐昊 天津大学

报告题目：Stochastic fluid dynamics driven by pseudo-differential noise —From Stratonovich sense to Marcus sense

摘要：This talk investigates the well-posedness of stochastic differential equations driven by pseudo-differential noise within the frameworks of Stratonovich and Marcus integrals. The core objective is to elucidate the cancellation property of pseudo-differential noise,which not only contributes to understanding the intrinsic singularity of the equation but is also crucial for proving the existence and uniqueness of solutions.

10、黎怀谦 天津大学

报告题目：Dimension-free Maz'ya--Shaposhnikova formulas on integer lattices

摘要：We study Besov spaces on the integer lattice $\mathbb{Z}^d$,defined via the heat semigroup of the discrete Laplacian. We prove a dimension-free Maz'ya–Shaposhnikova formula for their seminorms and derive limiting results in $\ell^p(\mathbb{Z}^d)$ for the fractional discrete Laplacian for all $1\leq p<\infty$. Our analysis employs approximation techniques within $\ell^p$ norms,which avoids classical Fourier-analytic tools and is well-adapted to this discrete context.