“中南概率日”学术报告
发布时间：2023/04/27
报告时间：4月28日上午
报告地点：数理楼135
报告一.吴昊 清华大学
报告题目：Connection probabilities for 2D critical lattice models
摘要：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.
报告人简介：
吴昊，2009年本科毕业于清华大学数学系，2013年博士毕业于法国巴黎十一大；2013-2017年，先后在美国麻省理工与瑞士日内瓦大学做博士后；2017年，被聘为清华大学长聘教授。吴昊主要研究随机过程Schramm Loewner Evolution、高斯自由场与伊辛模型等经典统计物理模型。主要代表作包含平面统计物理模型连通概率系列工作等。
报告题目：Scaling limit of a long-range random walk in time-correlated random environment
摘要：This paper concerns a long-range random walk in random environment in dimension 1 + 1, where the environmental disorder is independent in space but correlated in time. We prove that the rescaled partition function converges weakly to the Stratonovich solution of a fractional stochastic heat equation with multiplicative Gaussian noise which is white in space and colored in time. This is a joint work with Guanglin Rang and Meng Wang.
报告人简介：
宋健，山东大学数学与交叉科学研究中心教授。主要研究方向为随机偏微分方程、统计物理模型、随机矩阵以及随机分析及其应用等。
报告三.肖惠 中国科学院
报告题目:Precise Large deviations for the coefficients of random walks on the general linear group
摘要：Consider a sequence $(g_n)_{n\geq 1}$ of independent and identically distributed random matrices and the left random walk $G_n : = g_n \ldots g_1$ on the general linear group $GL(d, \mathbb R)$. Under suitable conditions, we establish Bahadur-Rao-Petrov type large deviation expansions for the coefficients $\langle f, G_n v \rangle$ of the product $G_n$, where $v \in \mathbb R^d$ and $f \in (\mathbb R^d)^*$. In particular, we obtain an explicit rate function in the large deviation principle, thus improving significantly the known large deviation bounds. Moreover, we prove local limit theorems with large deviations for the coefficients, and large deviation expansions under Cram\'er's change of probability measure. For the proofs we establish the H\"older regularity of the invariant measure of the Markov chain $(\mathbb R G_n v)$ under the changed probability, which is of independent interest. Joint work with I. Grama and Q. Liu.
肖惠，中国科学院数学与系统科学研究院优秀青年副研究员。2020年博士毕业于法国南部列塔尼大学，2020年到2023年在德国希尔德斯海姆大学做博士后。主要研究方向为随机矩阵乘积、（分枝）随机游动等。相关论文发表（含接受发表）在J. Eur. Math. Soc.，Ann. Probab.，Ergodic Theory Dynam. Systems，Ann. Inst. Henri Poincaré Probab. Stat.，Stochastic Process. Appl.，J. Differential Equations等
报告题目：Entropy Method in the Mean Field Limit Problem
摘要：Entropy/energy based methods have been shown useful in the mean field limit/propagation of chaos of large systems of interacting particles. We will present recent progress in this direction and in particular the long time convergence result based on entropy/Fisher information.
报告人简介：
王振富，2012年本科毕业于南京大学，2017年获美国马里兰大学数学博士学位，博士导师为Pierre-Emmanuel Jabin。2017年7月到2020年6月在美国宾夕法尼亚大学从事博士后研究工作。2020年10月入职北京大学，现任北京国际数学研究中心助理教授、研究员。主要研究领域为交互粒子系统的平均场极限和动理学方程的分析。
报告五.鲍建海 天津大学
报告题目：Exponential ergodicity for damping Hamiltonian dynamics with state-dependent and non-local collisions
摘要：In this talk, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov processes while is very popular in numerous modelling situations including stochastic algorithms. The approach adopted in this talk is based on a combination of the refined basic coupling and the refined reflection coupling for non-local operators. In a certain sense, the main result developed in the present talk is a continuation of the counterpart in our previous work on exponential ergodicity of stochastic Hamiltonian systems with Levy noises. This talk is based on a joint work with Jian Wang.
报告人简介：
鲍建海，现任职于天津大学应用数学中心。2013年01月获英国斯旺西大学博士学位；2012年9月-2013年8月在美国韦恩州立大学从事Research Fellow；2017年1月-2019年12月在英国斯旺西大学从事博士后研究；2013年9月-2020年6月，在中南大学数学与统计学院工作。
报告题目：McKean-Vlasov SDEs with Sigularity in Distribution Variable and Distribution Dependent Diffusion
摘要：The well-posedness for SDEs with singularities in both space and distribution variables is derived, where the drift is bounded and Lipschitz continuous under weighted variation distance and the diffusion is allowed to be Lipschitz continuous under $L^\eta$($\eta\in(0,1]$)-Wasserstein distance. Moreover, the regularity estimate is provided when the drift is bounded and Lipschitz continuous under total variation distance.
报告人简介：
黄兴，2017年博士毕业北京师范大学概率论与数理统计专业，师从王凤雨教授，现为天津大学应用数学中心副教授。研究方向：随机分析。最近关注分布依赖的随机微分方程的解的适定性，混沌传播现象和分布性质如正则性估计和Harnack不等式等。
报告时间：4月28日下午14:40-15:40
报告地点：数理楼135
报告七.翟建梁中国科学技术大学
报告题目：Irreducibility of SPDEs driven by pure jump noise
摘要：The irreducibility is fundamental for the study of ergodicity of stochastic dynamical systems. In the literature, there are very few results on the irreducibility of stochastic partial differential equations (SPDEs) and stochastic differential equations (SDEs) driven by pure jump noise. The existing methods on this topic are basically along the same lines as that for the Gaussian case. They heavily rely on the fact that the driving noises are additive type and more or less in the class of stable processes. The use of such methods to deal with the case of other types of additive pure jump noises appears to be unclear, let alone the case of multiplicative noises. In this paper, we develop a new, effective method to obtain the irreducibility of SPDEs and SDEs driven by multiplicative pure jump noise. The conditions placed on the coefficients and the driving noise are very mild, and in some sense they are necessary and sufficient. This leads to not only significantly improving all of the results in the literature, but also to new irreducibility results of a much larger class of equations driven by pure jump noise with much weaker requirements than those treatable by the known methods. As a result, we are able to apply the main results to SPDEs with locally monotone coefficients, SPDEs/SDEs with singular coefficients, nonlinear Schrodinger equations, Euler equations etc. We emphasize that under our setting the driving noises could be compound Poisson processes, even allowed to be infinite dimensional. It is somehow surprising.
翟建梁，中国科学技术大学副教授，2010年获中国科学院数学与系统科学研究院博士。主要研究方向是Levy过程驱动的随机偏微分方程。已在 “J. Eur. Math. Soc.”、“J. Funct. Anal.”、“J. Math. Pures Appl.”等国际重要杂志发表论文三十余篇。
报告八.吴付科 华中科技大学
报告题目：Fast-slow-coupled stochastic functional differential equations
摘要：This paper focuses on two-time-scale coupled stochastic functional differential equations (SFDEs). The system under consideration has a slow component and a fast component. Both components depend on the segment process (an infinite dimension process) of the slow component. To overcome the difficulty due to the past dependence and the coupling of the segment process, such properties as the Hölder continuity and tightness on a space of continuous functions are investigated first for the segment process. In addition, it is also shown that the solution of a fixed-x equation depends continuously on the parameters. Then using the martingale problem formulation, an average principle is established by a direct averaging.
报告人简介：
吴付科，教授，博士生导师，2003年博士毕业于华中科技大学数学与统计学院。主要从事随机微分方程以及相关领域的研究，2011年入选教育部新世纪优秀人才支持计划，2012年入选华中科技大学“华中学者”，2014年获得基金委优秀青年基金资助，2015年获得湖北省自然科学二等奖，2017年获得英国皇家学会"牛顿高级学者"基金，《IET Control Theory & Applications》编委。近年来，发表论文80余篇，主持7项国家自然科学基金和一项教育部新世纪优秀人才基金，也主持过一项美国数学学会（AMS）基金。