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2008年10月
报告题目: On time change theory for symmetric Markov processes (I)
报 告 人: Prof. M. Fukushima (Professor Emeritus Osaka University, Japan)
时间地点: 2008年10月29日(星期三)下午2:30-3:30 思源楼703
摘       要: .
报告题目: Talagrand和对数Sobolev不等式
报 告 人: 吴黎明 教授,法国Blaise Pascal大学
时间地点: 2009年7月30日(星期四) 下午4:00-5:00 思源楼703室
摘       要: 我们通过实用的Lyapunov函数条件导出Talagrand不等式,并且在一定曲率条件下导出对数Sobolev不等式。最优运输映射起重要作用。
报告题目: On the Path to Extinction
报 告 人: Prof. Peter Jagers (Chalmers University of Technology, Sweden )
时间地点: 2008年10月27日(星期一)下午3:00-4:00 思源楼703
摘       要: Populations can die in many ways. Arguably the most basic is extinction for intrinsic reasons, by stably insufficient, i.e. subcritical, reproduction. We consider (moderately) large, subcritical general branching processes, and ask what the time T to extinction is. It turns out that T has two components, both carrying weight, a deterministic part, proportional to the logarithm of initial population size, and a random part, with an approximate Gumbel distribution. Then we consider the path to extinction. How large will the population be at time uT, 0<u<1? There is a beautiful limiting form, for large initial populations, of this as a stochastic process on the unit interval. For u close to one the approximation is bad but in the case of Markov branching process we can describe the last minutes before extinction. (Joint work with Fima Klebaner and Serik Sagitov, published in the Proc Nat. Acad. of US (2007) and J. Appl. Prob. (2007).
报告题目: Applications of a Markov mapping theorem
报 告 人: Prof. Thomas Kurtz (University of Wisconsin, USA )
时间地点: 2008年10月20日(星期一)下午3:00-4:00 思源楼703
摘       要: Conditions in terms of generators are given under which the image of a solution of a martingale problem with values in one metric space gives a solution of a martingale problem with values in the image space. Applications of the general result will be discussed including the equivalence between martingale problems and stochastic differential equations, and uniqueness for stochastic partial differential equations.
2008年9月
报告题目: Asymptotic regimes of the occupation scheme for multiplicative cascades
报 告 人: Dr. Jean Bertoin (Université Pierre et Marie Curie (Paris VI) )
时间地点: 2008年9月1日(星期一)下午3:40-4:40 思源楼703
摘       要: In the classical occupancy scheme, one considers a fixed discrete probability measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls independently at random in boxes labeled by ${\cal I}$, such that $p_i$ is the probability that a given ball falls into the box $i$. In this lecture, we are interested in asymptotic regimes of this scheme in the situation induced by a refining sequence $({\bf p}(k): k\in\N)$ of random probability measures which arise from some multiplicative cascade. Our motivation comes from the study of the asymptotic behavior of certain fragmentation chains.
报告题目: The continuous limit of large random planar maps
报 告 人: Dr. Jean-Francois Le Gall (Universite Paris XI and Institut universitaire de France)
时间地点: 2008年9月1日(星期一)下午2:30-3:30 思源楼703
摘       要: Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they also have significant geometrical applications. Random planar maps have been used in theoretical physics, where they serve as models of random geometry. Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all planar maps with n vertices in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power -1/4. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit can be written as a quotient space of the so-called Continuum Random Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels attached to the vertices of the CRT. This limiting random metric space, which is called the Brownian map, can be viewed as a "Brownian surface" in the same sense as Brownian motion is the limit of rescaled discrete paths. We show that the Brownian map is almost surely homeomorphic to the two-dimensional sphere, although it has Hausdorff dimension 4. Furthermore, we are able to give a complete description of the geodesics from a distinguished point (the root) of the Brownian map, and in particular of those points which are connected by more than one geodesic to the root. As a key tool, we use bijections between planar maps and various classes of labeled trees.
2008年7月
报告题目: Stein's method of exchangeable pairs with application to the Curie-Weiss Model
报 告 人: Prof. Qi-Man Shao (Hong Kong University of Science and Technology)
时间地点: 2008年7月2日(星期三)下午3:00-4:00 思源楼1013
摘       要:  
2008年4月
报告题目: On the existence of solutions for unidimensional BSDE's
报 告 人: Prof. Jean-Pierre Lepeltier (Universite du Maine, Le Mans, France)
时间地点: 2008年4月15日(星期二)下午4:00-5:00 晨兴数学中心110
摘       要:  
报告题目: Markov loops and fields
报 告 人: Prof. Yves Le Jan (巴黎南大学)
时间地点: 2008年4月14日(星期一)下午3:00-4:00 思源楼703室
摘       要: We will investigate Poisson processes of Markov loops and the associated occupation fields, which, in cetain cases are related to the free field. Renormalisation problems will be adressed in the case of Brownian loops.
2008年3月
报告题目: 树,随机游动, 聚合物
报 告 人: 施展教授(巴黎第六大学)
时间地点: 2008年3月24日(星期一)下午3:00-4:00 思源楼703室
摘       要: 演讲人简介:施展教授早年就读于中国科技大学(本科)和中科院应用数学研究所(研究生), 1986年负笈法国巴黎,师从著名概率学家Marc Yor, 1990年获博士学位,1998年任巴黎第六大学概率论实验室(Laboratoire de Probabilités et Modèles Aléatoires)教授至今。施展教授的研究范围包括布朗运动、随机游动、大偏差估计等随机过程的多个方面,详细情况请见其个人主页 http://www.proba.jussieu.fr/pageperso/zhan/.
2007年12月
报告题目: A survey of quantum stochastic calculus
报 告 人: Prof. Robin Hudson, Loughborough University, UK
时间地点: 2007年12月13日(星期四)下午4:00-5:00 思源楼703室
摘       要: Quantum stochastic calculus originated as a noncommutative generalisation of the Ito calculus of Brownian motion, motivated by the so-called Wiener-Segal isomorphism between the L^2 space of Wiener space and a Fock space. The Poisson process and other Levy processes are naturally incorporated into the same formalism. Ito corrections are related to quantum commutation relations. Generalisations and some applications in physics and in finance will be discussed.
报告题目: Quantum constrained systems and quantum graphs
报 告 人: Prof. Gianfausto Dell'antonio,University Roma I and SISSA
时间地点: 2007年12月13日(星期四)下午3:00-4:00 思源楼703室
摘       要: In this talk I will outline the theory of constrained systems in Quantum Mechanics as a multiscale process. I shall give some examples and use it as a framework for the study of the limit Schroedinger equation on quantum graphs.
2007年11月
报告题目: Feynman-Kac formula and ralated problem
报 告 人: 应坚刚 教授, 复旦大学数学学院
时间地点: 2007年11月26日(星期一)下午3:00-4:00 思源楼712室
摘       要:  
2007年10月
报告题目: Stochastic porous media and Dirichlet spaces
报 告 人: Prof. Byron Schmuland, University of Alberta, Canada
时间地点: 2007年10月22日(星期一)下午3:00-4:00 思源楼703室
摘       要: Recent developments in solving stochastic porous media equations benefit from the theory of Dirichlet forms. I show how some older results on extended Dirichlet space help to put these equations in the right context.
2007年9月
报告题目: Insurance Claims Modulated by a Hidden Marked Point Process
报 告 人: Prof. Robert J. Elliott, RBC Financial Group Professor of Finance,Haskayne School of Business,University of Calgary, Canada
时间地点: 2007年9月24日(星期一)下午4:00-5:00 思源楼703室
摘       要: Recently there has been interest in applying Markov modulated Poisson processes to ruin theory. Such a model can incorporate the impact states of the economy have on the surplus process. However, the Markov chain is often not directly observable in practice. We develop smoothing and filtering results when the hidden Markov chain influences both the timing and size of insurance claims. The parameters of the model are also estimated.
报告题目: Ageing in Meanfield Spin Glasses
报 告 人: Prof. Anton Bovier,德国Weierstrass Institute
时间地点: 2007年9月17日(星期一)下午3:00-4:00 思源楼703室
摘       要: .
2007年8月
报告题目: Probability and Fourier Transforms with Only Real Zeros
报 告 人: Prof. Wenbo Li, University of Delaware
时间地点: 2007年8月6日(星期一)上午10:00-11:00 思源楼703室
摘       要: The study of Fourier transforms with only real zeros has significant impacts in combinatorics, analysis and probability. We will provide several examples to demonstrate interplays with probability.
报告题目: On Independence Tests Based on Large Sample Correlation Matrices
报 告 人: Prof. Qi-Man Shao, Hong Kong University of Science and Technology,University of Oregon
时间地点: 2007年8月6日(星期一)上午9:00-10:00 思源楼703室
摘       要: Let X1, · · · ,Xn be a random sample from a p-dimensional population distribution. Assume that both p and n are large. To test whether the p-variates of the population are independent, Jiang (2004) uses the largest entry of the sample correlation matrix as a test statistic and the limiting distribution is the extreme distribution of type I. It is known that the rate of convergence to this type of extreme distribution is typically slow, of order of O(1/ log n). In this talk we will introduce a new test statistic which also has an extreme limiting distribution of type I but with a rate of convergence O((log n)3/n1/2). Other related problems will also be discussed. This talk is based on a joint work with W.D. Liu and Z.Y. Lin at Zhejiang University.
2007年7月
报告题目: Poinacre inequalities and Sobolev inequalities on differential forms over complete Riemannian manifolds
报 告 人: Prof. Xiang-Dong Li, University of Toulouse III, France
时间地点: 2007年7月27日(星期五)下午4:00-5:00 思源楼712室
摘       要: In this talk, I will present some of my recent work on Poincare inequalities and Sobolev inequalities on differential forms over complete Riemannian manifolds. As application, I present some $L^p$-estimates and existence theorems of the De Rham equation $d\omega=\alpha$ as well as some vanishing theorems of the $L^p$-cohomology on complete Riemannian manifolds.
报告题目: Schramm Loewner Evolutions and Generalizations
报 告 人: 关庆扬 助理研究员, Institute of Applied Mathematics, CAS Department of Mathematics,Imperial College London
时间地点: 2007年7月27日(星期五)下午3:00-4:00 思源楼712室
摘       要: Schramm Loewner Evolutions (SLE) are new methods to study conformal invariant curves or compact sets in two dimensional plane. In this talk we first give an introduction of this excited subject. Then we consider stochastic Loewner equation driven by more general L\'evy processes which give increasing clusters with trees-like structure. We show that the increasing clusters can be generated by c\'adl\'ag curves. We also introduce a new class of ordinary differential equations called $\alpha$-Loewner equation and define $\alpha$-SLE. The corresponding clusters have $\alpha$-self-similarity. We show the phase transition of $\alpha$-SLE at a critical coefficient $\theta=\theta_0(\alpha)$ analogous to the $\kappa=4$ phase transition in Brownian SLE. Finally we discuss the possibility of SLE in high dimensional cases.
报告题目: Potential theory of subordinate Brownian motions
报 告 人: 宋仁明 教授, University of Illinois at Urbana-Champaign
时间地点: 2007年7月23日(星期一)下午3:00-4:00 思源楼712室
摘  要: In this talk, I will give a survey of recent progress in the study of potential theory of subordinate Brownian motions.
2007年6月
报告题目: Rounded Data Analysis
报 告 人: 白志东院士, 东北师范大学
时间地点: 2007年6月4日(星期一)下午3:00-4:00 思源楼712室
摘  要:

Except categorical data, all continuous data need to be rounded when they are collected and recorded. The rounding errors definitely affect the accuracy of statistical inferences. In old days, it was not seriously treated because statistical problems are only met for small samples. However, with the wide application of advanced computers, statisticians are gradually facing to deal with data of large sample with large dimension. This forces our statisticians to pay serious attention to the rounding errors. It has been found in the literature that the usual t-test will reject the true null hypothesis with probability near to 1 when the sample size is large enough. This shows that we have to search new methodologies to deal with analysis of data when the observations are rounded and the rounding scale is large relatively to the sample size. In a series of my recent research works, we proposed some new methods to deal with the analysis of rounded data

.In this talk, I will revisit an example discussed by Dempster and Rubin in their 1982 JRSSB paper. We will show that Sheppard corrections are superior to the BRB corrections only for their example and generally not the case. Further, we propose a new method to find consistent and asymptotically normal estimates in rounded linear models. The new method is named as two-stage estimation
.

2007年4月
报告题目: Quantum Entanglement and Related Mathematics
报 告 人: Prof. Shao-Ming Fei, Bonn University,Capital Normal University
时间地点: 2007年4月12日(星期四)下午4:00-5:00 思源楼703室
摘  要: We introduce some basic concepts and recent developments in the theory of quantum entanglement used in quantum information processing and computation, in particular, the measure of entanglement, separability criteria, invariants and geometry of quantum states under local unitary transformations.
2007年1月
报告题目: Geometric Stochastic Analysis on Path Spaces
报 告 人: Prof. K. David Elworthy , Mathematics Institute, Warwick University, UK
时间地点: 2007年1月4日(星期四)下午4:00-5:00 思源楼703室
摘       要: An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with 'Brownian motion' measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on Rm and Malliavin calculus. For manifold valued paths the approach is to use 'It?o' maps of suitable stochastic differential equations as charts. 'Suitability' involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting 'Laplacian' are described. A theory for more general diffusion measures is also briefly indicated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one and two forms leading to a Hodge-Kodaira operator and decomposition for such forms. Finally there is a brief description of some related results for loop spaces.
2006年12月
报告题目: Hranack不等式的耦合方法
报 告 人: 王凤雨 教授 , 北京师范大学
时间地点: 2006年12月18日(星期一)下午4:00-5:00 思源楼703室
摘       要: .介绍使用耦合方法证明Harnack不等式的基本思想, 并应用于流形上具无下界曲率的扩散半群以及一类随机偏微分方程的研究。特别地,证明了一般随机porous medium 方程的几种超压缩性、强Feller性和热核估计等
2006年11月
报告题目: Expected Length of Minimum Spanning Tree
报 告 人: Prof. Wenbo Li, University of Delaware, USA
时间地点: 2006年11月13日(星期一)下午4:00-5:00 晨兴数学中心110
摘       要: An exact formula for the expected length of the minimum spanning tree of a connected graph with independent and identical edge distribution is given, which generalizes Steele's formula in the uniform case. For complete graph, the difference of expected lengths between exponential with rate one and uniform on the interval (0,1) is shown to be positive and of rate zeta(3)/n..
This is a joint work with Xinyi Zhang .
2006年10月
报告题目: Works of 2006 Fields Medalists in Probability
报 告 人: Prof. Wenbo Li, University of Delaware, USA
时间地点: 2006年10月23日(星期一)下午4:00-5:00 晨兴数学中心110
摘       要: The Fields Medals are the most important international prize in the world of mathematics.They are awarded by the International Mathematical Union (IMU) every four years at the ICM (International Congress of Mathematicians).Between two and four Fields Medals can be awarded at each ICM to only mathematicians below the age of 40 in order to encourage future endeavor..They are typically awarded for a body of work, rather than a single isolated achievement.
The winners of the Fields Medals awarded in ICM2006 are: Andrei Okounkov, Grigori Perelman, Terence Tao, and Wendelin Werner.Their works related to probability theory will be discussed in this talk, which should be understandable for a wide audience.I hope to convince the audience/students that they should learn as much probability as possible.Without any double, probability theory has became one of the most fascinating, fast growing and main stream mathematics.
2006年9月
报告题目: Local stochastic differential geometry: feeling the shape of a manifold with Brownian motion
报 告 人: Prof. Mark Pinsky, Department of Mathematics, Northwestern University, USA
时间地点: 2006年9月11日(星期一)下午3:00-4:00 思源楼703
摘       要: .
2006年7月
报告题目: Relativistic Diffusions and Schwarzschild Geometry
报 告 人: Prof. Jacques FRANCHI(University of Louis Pasteur, Strasbourg, France)
时间地点: 2006年7月21日(星期五)下午3:00-4:00 思源楼703
摘       要: This talk is based on a joint work with Yves LeJan. We shall develop Ito calculus for relativistic diffusions.
报告题目: Limiting Theorems Associated with Poisson-Dirichlet Distributions
报 告 人: Prof. Shui Feng(Dept. of Mathematics & Statistics, McMaster University, Canada)
时间地点: 2006年7月3日(星期一)下午3:00-4:00 思源楼703
摘       要: Some limiting theorems will be discussed for Poisson-Dirichlet distribution and its random sample. These results are motivated for the understanding of the exact meaning of large mutation rate.
2006年6月
报告题目: Gaussian Inequalities and Conjectures
报 告 人: Prof. Wenbo Li (University of Delaware, USA)
时间地点: 2006年6月12日(星期一)下午3:00-4:00 思源楼712
摘       要: ..
报告题目: Spectral Measure of Large Hankel, Markov and Toeplitz Matrices
报 告 人: Tiefeng Jiang (Associate Professor,School of Statistics, University of Minnesota, USA
时间地点: 2006年6月12日(星期一)下午3:00-4:00 思源楼712
摘       要: We study the limiting spectral measure of large symmetric random matrices. This includes the asymptotic behavior of properly scaled eigenvalues of Hankel, Markov and Toeplitz matrices. This solves three unsolved random matrix problems. It is the joint work with W. Bryc and A. Dembo.
2006年5月
报告题目: Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains
报 告 人: 宋仁明 教授 (University of Illinois at Urbana-Champaign)
时间地点: 2006年5月18日(星期四)下午3:00-4:00 晨兴数学中心110
摘       要: The notion of intrinsic ultracontractivity, introduced by Davies and Simon for symmetric semigroups, is a very important concept and has been studied extensively. However, it seems that, up to now, no one has introduced the concept of intrinsic ultracontractivity for non-symmetric semigroups. In this talk, we first introduce the notion of intrinsic ultracontractivity for non-symmetric semigroups. We will show that the Dirichlet semigroups of non-symmetric diffusion semigroups are intrinsic ultracontractive on very rough domains.
2006年3月
报告题目: Gibbs测度的谱隙估计
报 告 人: 吴黎明 教授(法国Blaise Pascal大学, 中科院数学与系统科学研究院(国家杰出青年B类获得者))
时间地点: 2006年3月30日(星期四)下午3:00-4:00
摘       要: 利用Bocher公式的思想,将Ligget M-ε定理推广到连续spin空间情形,由此得到Gibbs场谱隙的sharp估计。作为应用,讨论SN-模型和离散P(φ)d模型
2005年12月
报告题目: Zeros of Random Polynomials
报 告 人: 方诗赞 教授(法国Bourgogne大学)
时间地点: 2005年12月19日(星期一)下午3:00-4:40 晨兴110
摘       要: We shall give a survey on our works, some of them being done jointly with Tusheng Zhang and Peter Imkeller, about stochastic differential equations beyond non Lipschitz conditions..
报告题目: Zeros of Random Polynomials
报 告 人: 李文博 教授(美国Univ. of Delaware )
时间地点: 2005年12月12日(星期一)下午4:00-5:00 712
摘       要: There is a long history on the study of the number of real zeros of a random polynomial whose coefficients are independent, identically distributed random variables. We first give an overview and then present new representations and bounds for the decay (positivity) exponent of the probability that a random polynomial of large even degree has no real zero. Similar results for multi-variable random polynomial are also given. Connections with Gaussian processes and various related probability estimates are discussed..
2005年9月
报告题目: The Hypergroup Property and Representation of MarkovKernels
报 告 人: Professor Dominique Bakry(University Paul Sabatier, France)
时间地点: 2005年9月12日(星期一)4:00-5:00 703
摘       要: ..
报告题目: Rough Path Theory
报 告 人: Professor Terry Lyons(Fellow of Royal Society Mathematical Institute, University of Oxford)
时间地点: 2005年9月5日(星期一)4:00-5:00 712
摘       要: The theory of rough paths as developed by the Author (and several others such as Hambly, Ledoux, Coutin, Qian, Friz, ...) aims to extend the calculus of differential equations to contexts that are far from differentiable. The methodology has had a significant impact on classical questions from probability, but also has connection with topology and geometry. The methodology extends classical and Ito calculus to many new types of stochastic forcing outside the classical semi-martingale setting directly.
The key to this theory is to answer the question - when do two controls produce similar responses. This is also a core question for the problem for multi-scale analysis where one needs to summarise small scale behaviour in a way that large scale responses can be predicted from the summarised information. The question can be translated into one asking that one characterises the continuity properties of the Ito map. This is indeed possible and the Universal Limit theorem proves the (uniform) continuity of the map taking the forcing control to response for a wide class of metrics on smooth paths - and the completions of the space under these metrics give the so called rough paths - giving insight into the control problem. There are many immediate consequences of this universal limit theorem within probability theory.
The approach is quite structured, and allows one to give a top down analysis of a control in terms of a sequence of algebraic coefficients we call the signature of the control (which have similarity to a child's precis of a complicated text by a simpler one and are a non-commutative analogue of Fourier coefficients) with refinements giving more accurate information about the control. Hambly and Lyons recently proved that this "signature" of a control completely characterises a finite length control up to the appropriate null sets..
2005年6月
报告题目: Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature
报 告 人: Prof. Anton Thalmaie(University of Poitiers, France )
时间地点: 2005.6.3 2:30-3:30pm(数学院思源楼703)
摘       要: The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) has attracted a lot of attention. In particular, for Cartan-Hadamard manifolds the necessity of lower curvature bounds is still an open problem. We discuss examples of such manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case..
2005年4月
报告题目: Weak and Strong Sobolev Spaces over path spaces
报 告 人: Dr. Xue-Mei Li (Nottingham Trent University)
时间地点: 2005.4.4 4:00-5:00pm(数学院思源楼712)
摘       要: The spaces of paths provide a particularly interesting family of infinite dimensional spaces for their connection with physics and stochastic analysis. One of the most imoportant tools in analysis in Sobolev calculus. Here we investgiate the property of the basic differential operator d. Although there had been a surgy of results in this direction, including the establishement of integration by parts formulae and Sobolev inequalities, the uniqueness of d had not been investigated. For example on the Wiener space, spaces of continuous paths on Euclidean spaces, the closure of d has a characterisation. We ask whether the same works on path spaces and give a uniqueness result. This is joint work with K. D. Elworthy..
2005年3月
报告题目: Ergodic properties of the 2D Navier-Stokes equations I
报 告 人: Dr. Martin Hairer (Warwick University)
时间地点: 2005.3.21 4:00-5:00pm(晨兴数学中心110 )
摘       要: In this talk, I will present some recent developments in the study of the 2D stochastic Navier-Stokes equations driven by translation-invariant Gaussian noise. I will give a geometric characterisation of a class of noises that leads to ergodicity and present the main ideas of the proof of this statement. This talk is based on a joint work with J. Mattingly..
报告题目: 本质谱半径的变分公式在Navier—Stoke方程中的应用
报 告 人: 吴黎明教授(武汉大学、法国Blaise Pascal 大学)
时间地点: 2005.3.11 4:00-5:00pm
摘       要: 对二维Navier—Stokes方程的离散扰动构成的马氏过程,我们利用本质谱半径的变分公式得到它的非质谱半径估计:

⑴在扰动是高斯且非退化情形下,证明了它在加权空间bvB上的本质半径为0,从而得到大偏差原理。

⑵在扰动是有界的情形下,证明了它在Lipschitz函数空间CLip上有谱隙。

这些结果是二维Navier—Stokes方程随机扰动已知结果的推广和精细。

2005年1月
报告题目: INTEGRAL STOCHASTICEQUATION AND ITS ASYMPTOTIC BEHAVIOUR
报 告 人: Professor Hisao Fujita Yashima(Department of MathematicsUniversity of Turin, ITALY)
时间地点: 2005.1.10 3:00-4:00pm
摘       要: We consider first a linear integral stochastic equation (Volterra-Ito equa-tion) with values in Rn. Under a suitable ( and reasonably general ) condition we prove the uniform boundedness of E(|X(t)|2). Under the same condition we prove also, for this equation, the existence of the (unique) invariant measure in a suitably constructed Hilbert space. This Hilbert spacecontains the "present" and the "past"of the process. The most part of this result can be generalized to a class ofnon-linear integral stocastic equations..
2004年12月
报告题目: Stein's method and discreteprobability approximations
报 告 人: 夏爱华 教授(Department of Mathematics and Statistics Univ. of Melbourne, VIC 3010, Australia)
时间地点: 2004.12.20 5:00-6:00pm
摘       要: We want to approximate for , where is a sum of independent (or weakly independent) integer-valued random variables with finite second moments. This talk covers the following two cases.
Case 1: Majority of are small, for example, W counts the number of occurrences of certain rare events
Case 2: Majority of are relatively large, then the distribution of should behave like a "discrete normal".
The first case are approximated by Poisson or a modified Poisson such as compound Poisson, Poisson signed measures with errors of approximation estimated by Stein-Chen method. Barbour's probabilistic interpretation of Stein-Chen method for estimating the error of Poisson approximations not only paved a way for investigating Poisson process, but also provide an opportunity for studying other approximations. Xia (1999) give a pure probabilistic proof of stein bound for Poisson approximation, and the case of approximations by general distributions on the non-negative integers by Brown and Xia (2001). The method of Brown and Xia apply to a very large class of approximation distributions on the non-negative integers, including Poisson, binomial, negative binomial as well as a natural class for high-order approximations by probability
distributions rather than signed measures. This offers a comprehensive solution of case 1.
In terms of case 2, Goldstein and xia (2004) introduce a new family of discrete distributions and provided a discrete version of Berry-Esseen theorem showing how members of family approximate the distribution of in total variation ..
2004年11月
报告题目: Potential theory of specialsubordinators and subordinate killed Brownian motions
报 告 人: 宋仁明 教授(Dept. of Math., Univ. of Illinois at Urbana-Champaign)
时间地点: 2004.11.22 5:00-6:00pm
摘       要: In this talk we will first introduce a class of subordinators called special subordinators. The class of subordinators is very large and it includes all the subordinators whose Laplace exponents are complete Bernstein functions, so it includes all the stable subordinators. We then present the potential theory of special subordinators. We then present results on
the potential theory of processes obtained by subordinating a killed Brownian motion in a bounded Lipschitz domain D by special subordinators. We will show that there is a one-to-one correspondence between the nonnegative harmonic functions of the killed Brownian motion and of the subordinate killed Brownian motion. By using this correspondence, we will show that both the Martin boundary and minimal Martin boundary of the subordiante killed Brownian motion coincide with the Euclidean boundary of D. We will also establish a boundary Harnack principle for the positive harmonic functions of subordinate killed Brownian motions..
报告题目: Stochastic Models of Economic Optimization
报 告 人: 陈木法 院士(北京师范大学数学科学学院)
时间地点: 2004.11.08 5:00-6:00pm
摘       要: Some stochastic models of economic optimization are discussed in this paper. Due to the value in practice, the models are quite attractive. But our knowledge on them is still very limited, some fundamental problems remain open.
We begin with a short review of the study on some global economic models (or economy in large scale), the well-known input-output method and L.K. Hua's fundamental theorem for the stability of economy. Then, we show that it is necessary to study the stochastic models. A collapse theorem for a non-controlling stochastic economic system is introduced. In the analysis of the system, the products of random matrices play a critical role. Especially, the first eigenvalue, the corresponding eigenfunctions and an ergodic theorem of Markov chains play a nice role here. Partial proofs are included. Some challenge open problems are also mentioned..
2004年10月
报告题目: Random Graphs andStochastic Complex Networks
报 告 人: 马志明 院士(中科院数学院应用数学所)
时间地点: 2004.10.25 5:00-6:00pm
摘       要: The last few years have witnessed a tremendous activity devoted to the characterization and understanding of growing networks and graphs in various systems such as the Internet, social networks, and biological networks. In this talk we shall first briefly review some results on the classical random graphs. And then we review some recent discussion on the study of stochastic complex networks arising from the real world. Among many hundreds of interesting papers we shall introduce in particular:
(i) "ten leading questions for network research" suggested by physicists in this area including A.L. Barabasi, J.F.F. Mendes el al;
(ii) view points of some mathemati- cians including D. Aldous, B. Bollobos and R. Durrett..
报告题目: An Invitation to Random Matrices
报 告 人: 骆顺龙 研究员(中科院数学院应用数学所)
时间地点: 2004.10.11 5:00-6:00pm
摘       要: Random matrices have their origin in multivariate statisticalanalysis (Pao-lu Hsu (许宝禄), Wishart, 1930s). They were in-troduced into physics by Wigner in the 1950s to model complex systems in nuclear physics. The basic idea underlying this approach is the assumption that the local statistical behavior of the energy levels of suciently complicated systems can be stim-ulated by that of the eigenvalues of random atrices. Also in the 1950s, Loo-keng Hua(华罗庚) calculated the volumes of various classical domains and the othogonal (unitary) groups which are important matrix spaces and thus paved the way for introducing natural probability distributions on such spaces. In the 1970s,Montgomery and Dyson discovered that the pair correlations of the zeros of the Riemann zeta function seems behave like that of the eigenvalues of certain random matrices. Around 1990s, Voiculescu discovered that asymptotically random matrices can serve as a natural model for the newly emerging held of free prob-ability. Now random matrices have developed into a profound enterprise which has fascinating and mysterious relations with,and deep applications in, statistics and probability, physics, number theory, operator algebras, and many others. This talk will provide a brief account of some stories around random matrices from a historical perspective.
2004年9月
报告题目: Vicious Walks, Schur Functions and Random Matrix Theory
报 告 人: Professor Makoto KATORI (Chuo University, Tokyo, Japan)
时间地点: 2004.9.27 5:00-6:00pm
摘       要: The system of one-dimensional symmetric simple random walks,in which none of walkers have met others in a given time period,is called the vicious walker model. It was introduced by Michael Fisher and applications of the model to various wetting and melting phenomena were described in his Boltzmann medal lecture in 1984.In the present talk, I will try to explain interesting connections among representation theory, random matrix theory and probability theory, using this simple diffusion particle system: Each vicious walk of N walkers is represented by an N-tuple of nonintersecting lattice paths on the spatio-temporal plane. We show that there is established a simple bijection between nonintersecting lattice paths and semistandard Young tableaux. Based on this bijection and some knowledge of symmetric polynomials called the Schur functions, we can give a determinantal expression to the partition function of vicious walks, which is regarded as a special case of Karlin-McGregor formula in probability theory. Due to a property of Schur function, we can take the diffusion scaling limit of the vicious walks and define a noncolliding system of Brownian particles. This diffusion process solves the stochastic differential equations with the drift terms acting as the repulsive two-body forces proportional to the inverse of distances between particles, and thus it is idetified with Dyson's Brownian motion model. In other words, the obtained noncolliding system of Brownian particles is equivalent in distribution with the eigenvalue process of a Hermitian matrix-valued process..
报告题目: Recent development on interacting diffusion systems
报 告 人: Professor Tokuzo Shiga (Tokyo Institute ofTechnology)
时间地点: 2004.9.13 5:00-6:00pm
摘       要: ..
报告题目: A Relativistic Diffusion
报 告 人: Prof. Yves Le Jan ((University Paris - Sud, Orsay, RANCE 巴黎11大概率统计实验室主任)
时间地点: 2004.9.13 4:00-5:00pm
摘       要: The relativistic motion of a point mass with Brownian 4-velocity is defined in the general relativistic framework, extending a previous work of Dudley done in the context of special relativity. The exemple of the Schwartzschild space is studied in some detail especially to determine the asymptotic behaviour of the motion..
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