2018北京大学概率组研究生迎新会手册

单位：北京大学数学科学学院

时间：2018年9月17日，13：30—17：30

地点：北京大学理科一号楼1560

组织者：孙振尧

会议日程

题目与摘要

Hopf 模型的改进

报告人：白成，北京大学硕士生，导师刘勇

摘要：湍流问题的刻画是流体力学中重要的问题之一，而其一个中心问题是，分析描述粘性不可压缩流体动量守恒的运动方程 Navier-Stokes 方程解的存在性和光滑性。但对该方程的分析较为困难，因此Hopf提出一类简单的模型用于描述湍流的运动特征，但由于其对各维数之间做了独立性假设，因此和实际的湍流运动有着本质的偏差。本文在 Hopf 模型的基础上，加入扰动项和随机扰动项，改进了原模型。本文介绍了原 Hopf 模型的各项结果，并分析了其解的存在唯一性，同时对新模型分析了解的存在唯一性，以及其不变分布等相关的概率性质。

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《连续时间Galton—Watson tree上的极限定理》读书报告

报告人：崔词茗，北京大学硕士生，导师任艳霞

摘要：考虑一个基于连续时间超临界Galton—Watson tree上的粒子演化问题。粒子演化服从一个连续时间分支过程，其子代位置取决于母代位置和子代数量。本文通过构造一个和原Markov过程有相同生成元的辅助过程，得到在经验测度下t时刻存活个体数量相关的大数律，并分析了其与原始演化问题的联系。最后，利用所得结果，研究了几个具体特例，如中心极限定理、分裂扩散问题、细胞老化、分支Levy过程等。

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对角OU方程的Cylindrical 右连左极修正性质

报告人：杜燕，北京大学硕士生，导师刘勇

摘要：本文主要研究有着α -stable 过程扰动的对角OU方程的Cylindrical 右连左极修正以 及V-cylindrical右连左极修正性质.运用Basseo'Connor A and Rosiński J [2]中定理4.3,我们提出了:存在Cylindrical右连左极修正以及V-cylindrical 右连左极的又一充分条件.

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Anomalous contribution and fluctuation relations of singular perturbed diffusion process

报告人：金晓，北京大学博士生，导师葛颢

摘要：This work considers general diffusion processes taking place on widely separated time scales, and presents a measure-theoretic approach to analyze various general thermodynamic functional along sample paths, which satisfy fluctuation theorems. We find out that the limit of these functionals with odd and even variables under first-order and second-order singular perturbations fails to be the one directly defined on the limiting diffusion processes after the rapid dimensions have been eliminated. Their difference is called an anomalous term, which turns out to be an exponential martingale in all cases under consideration and satisfies the fluctuation theorems. Sufficient and necessary conditions for the vanishing of these anomalous terms are rigorously derived. Physical applications have also been included, especially for the second-order diffusion processes towards overdamping limit.

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多重 Ito 积分的 LDP 性质

报告人：黄翔宇，北京大学博士生，导师刘勇

摘要：定义多重 Ito 积分 $I_m(h) = \int_{T^m}h(t_1,\dots,t_m)W(d_1)\dots W(dt_m)$，我们希望研究其LDP 性质，以此来分析随机过程 $Y_t = \epsilon^{m/2}I_m(h_t)$ 的收敛性质。E. M. Wolf, D. Nualart, V. P. Abreu 于 1985 年通过收敛版本的压缩原理结合 H. Meyer 公式，得到了多重 Ito 积分的 LDP 性质。而在 1998 年，M. Boue, P. Dupuis 提出了变分表示公式，并由此得到了 Lipschitz 连续的随机微分方程强解的 LP 性质。本文希望利用变分表示公式，来研究多重 Ito 积分的 LP 性质，从另一个角度来阐述多重 Ito 积分的 LDP 性质。

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一族带指数扰动的马氏链的亚稳态性

报告人：高家兴，北京大学博士生，导师蒋达权

摘要：介绍陈大岳，冯建峰和钱敏平三位老师的文章《一族带指数扰动的马氏链的亚稳态性》，该文章考虑有限状态空间上一可约马氏链及其指数扰动的不可约马氏链族，是Freidlin-Wentzell理论的推广。证明了一大类马氏链族都具有亚稳态性，分析了高阶吸引子，吸引域以及所形成的金字塔形结构。

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The reversibility of stochastic reaction model and its relationship with multistability

报告人：李佑铭，北京大学博士生，导师蒋达权

摘要：We will firstly give an easily verifiable necessary and sufficient condition for the stochastic reaction model to satisfies Kolmogorov's Criterion under some structural conditions. Then we will discuss the relationship between the reversibility of stochastic reaction model and “quasi-potential”, which plays a key role in the corresponding Freidlin-Wentzell theory. As a result, a subclass of chemical system that exhibit multistability will be provided, and the transitions between different steady states can be explicitly described.

Finally, we will consider how to extend these results to general case, to be specific, we will drop the technical conditions and extend the results to the stochastic reaction model with non-mass action kinetics.

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The complete convergence theorem for coexistence threshold voter model on $T_d$

报告人：木英心，北京大学博士生，导师陈大岳

摘要：We consider threshold voter model $\eta_t$ on the homogeneous tree $T_d$. Firstly, we prove that except for the one-dimensional case, coexistence occurs. i.e. there exists a non-trivial invariant measure $\nu_{1/2}$. Secondly, we prove that $\eta_t$ converges to a linear combination of $ν_{1/2}$, $δ_0$ and $δ_1$ for any initial measure.

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The random walk in random environment

报告人：钱力，北京大学博士生，导师陈大岳

摘要：The random walk in deterministic environment is well-known and harvested abundantly, while the random walk in random environment is a newer branch. The random environment has various definitions. So far now I am interested in the environment whose transition probability is stochastic. Firstly, I would exhibite some fundamental results on the graph of $\mathbb{Z}$ and some trees. The work on $\mathbb{Z}$ is the earliest since 1970s, and the work on trees is younger and the process routine follows the former one sometimes. The trees studied at most are regular tree $\mathbb{T_d}$ and Galton-Watson tree. Secondly, I would give a brief statement about my research scheme.

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Limit theorems for a class of critical superprocesses with stable branching

报告人：孙振尧，北京大学博士生，导师任艳霞

摘要：We consider a critical superprocess $\{X;\mathbf P_\mu\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index $\gamma_0 > 1$. We first show that, under some conditions, $\mathbf P_{\mu}(\|X_t\|\neq 0)$ converges to $0$ as $t\to \infty$ and is regularly varying with index $(\gamma_0-1)^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t(f);\mathbf P_\mu(\cdot|\|X_t\|\neq 0)\}$, after appropriate rescaling, converges weakly to a positive random variable $\mathbf z^{(\gamma_0-1)}$ with Laplace transform

This is a joint work with Yan-xia Ren and Renming Song.

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CLT for Super-OU Processes with stable Branching

报告人：赵建杰，北京大学博士生，导师任艳霞

摘要：Consider a super-OU-process $\{X_t, t\geq 0\}$ with branching mechanism $\psi(z)=-\alpha z +z^{1+\beta}$ with $\alpha >0$ and $\beta\in (0,1)$. For each tesing function $f$ with polynomial growth, denoted by $\kappa(f)$ the order of $f$ in term of the spectral decomposition of $f$ with respect to the mean semigroup of $X$. We establish some spatial central limit theorem for $\langle f,X_t\rangle$ under three different regimes: In the small branching case $\alpha \beta <\kappa(f)b(1+\beta)$, conditioned on no-extinction, $\langle f,X_t\rangle/\|X_t\|^{\frac{1}{1+\beta}}$ convergences weakly; In the critical branching case $\alpha \beta =\kappa(f)b(1+\beta)$, conditioned on no-extinction, $\langle f,X_t\rangle/(t\|X_t\|)^{\frac{1}{1+\beta}}$ convergences weakly; In the large branching case $\alpha \beta >\kappa(f)b(1+\beta)$, conditioned on no-extinction, $e^{-(\alpha-\kappa(f)b)t}\langle f,X_t\rangle$ convergences a.s. and in $L^{1+\gamma}$ with $\gamma \in (0,\beta)$. This is an ongoing work with Yan-xia Ren, Renming Song and Zhenyao sun.

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The Limiting Behavior of the FTASEP with Product Bernoulli Initial Distribution

报告人：赵林杰，北京大学博士生，导师陈大岳

摘要：We study the facilitated totally asymmetric exclusion process on the one dimensional integer lattice. We investigate the invariant measures and the limiting behavior of the process. We mainly derive the limiting distribution of the process when the initial distribution is the Bernoulli product measure with density $1/2$. We also prove that in the low density regime, the system finally converges to an absorbing state.