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Supercritical superprocesses: Proper normalization and non-degenerate strong limit

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Science China Mathematics 2019年第8期62卷 1519-1552页

作者： Yan-Xia Ren renming song Rui Zhang LMAM School of Mathematical Sciences & Center for Statistical Science Peking University Beijing 100871 China Department of Mathematics University of Illinois Urbana IL 61801 USA School of Mathematical Sciences Nankai University Tianjin 300071 China School of Mathematical Sciences Capital Normal University Beijing 100048 China

Suppose that X ={Xt, t≥0;Pμ} is a supercritical superprocess in a locally compact separable metric space E. Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ0 of the generator of the mean semigroup of X. Then Mt := e-λ0t〈φ0,Xt〉 is a positive martingale. Let M∞ be the limit of Mt. It is known(see Liu et al.(2009)) that M∞ is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γt on [0,∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E,lim/t→∞γt〈φ0,Xt〉=W, a.s.-Pμ.We also give the almost sure limit of γt〈f, Xt〉for a class of general test functions f.

关键词： superprocesses Seneta-Heyde norming non-degenerate strong limit martingales

Suppression of explosion by polynomial noise for nonlinear differential systems

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Science China(Information Sciences) 2018年第7期61卷 161-171页

作者： Lichao FENG Shoumei LI renming song Yemo LI College of Science North China University of Science and Technology College of Applied Sciences Beijing University of Technology School of Mathematics and Research Center for Complex Systems and Network SciencesSoutheast University Department of Mathematics University of IllinoisUrbana

In this paper, we study the problem of the suppression of explosion by noise for nonlinear non-autonomous differential systems. For a deterministic non-autonomous differential system dx（t） =f(x（t）, t)dt, which can explode at a finite time, we introduce polynomial noise and study the perturbed system dx（t） = f(x（t）, t)dt + h（t）;|x（t）|βAx（t）d B（t）. We demonstrate that the polynomial noise can not only guarantee the existence and uniqueness of the global solution for the perturbed system, but can also make almost every path of the global solution grow at most with a certain general rate and even decay with a certain general rate（including super-exponential, exponential, and polynomial rates） under specific weak conditions.

关键词： explosion suppression noise non-autonomous differential system growth with general rate decay with general rate general polynomial growth condition

L log L criterion for a class of multitype superdiffusions with non-local branching mechanisms

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Science China Mathematics 2019年第8期62卷 1439-1462页

作者： Zhen-Qing Chen Yan-Xia Ren renming song Department of Mathematics University of Washington Seattle WA 98195 USA LMAM School of Mathematical Sciences & Center for Statistical SciencePeking University Beijing 100871 China Department of Mathematics University of Illinois Urbana IL 61801 USA

In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with nonlocal branching mechanisms under a martingale change of measure. As an application of this decomposition,we obtain a necessary and sufficient condition(called the L log L criterion) for the limit of the fundamental martingale to be non-degenerate. This result complements the related results obtained in Kyprianou et al.(2012),Kyprianou and Murillo-Salas(2013) and Liu et al.(2009) for superprocesses with purely local branching mechanisms and in Kyprianou and Palau(2018) for super Markov chains.

关键词： multitype superdiffusion non-local branching mechanism switched diffusion spine decomposition martingale

Uniform boundary Harnack principle for rotationally symmetric Lvy processes in general open sets

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Science China Mathematics 2012年第11期55卷 2317-2333页

作者： KIM Panki song renming VONDRACEK Zoran Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University Seoul 151-747 Korea Department of Mathematics University of Illinois Urbana IL 61801 USA Department of Mathematics University of Zagreb Zagreb Croatia

In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous Levy processes.

关键词： Levy processes subordinate Brownian motion harmonic functions boundary Harnack principle Poisson kernel

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一些常见分布族的反集中函数

数学理论与应用 2024年第1期44卷 1-15页

作者：胡泽春宋仁明谭渊四川大学数学学院成都610065 伊利诺伊大学厄巴纳香槟分校数学系 IL61801

假定{X_(α)}为一族服从某类分布的随机变量,具有有限期望E[X_(α)]和有限方差Var(X_(α)),其中α为一参数.受Hollom和Portier的论文(arXiv:2306.07811v1)的启发,在本文中我们考虑反集中函数(0,∞)∋y→inf_(α)P(|X_(α)-E[X_(α)]|≥y√Var(X_(α))),并给出其清晰表示.我们将证明,对于某些常见分布族,包括均匀分布、指数分布、非退化高斯分布和学生t分布,反集中函数不恒为零,这表明相应随机变量族具有某种反集中性质;然而对另外一些常见分布族,包括二项分布、泊松分布、负二项分布、超几何分布、伽马分布、帕雷托分布、威布尔分布、对数正态分布和贝塔分布,反集中函数恒为零.

关键词：分布测度反集中

Quasi-stationarity and quasi-ergodicity of general Markov processes

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Science China Mathematics 2014年第10期57卷 2013-2024页

作者： ZHANG JunFei LI ShouMei song renming College of Applied Sciences Beijing University of Technology Department of Mathematics University of Illinois

In this paper,we study the quasi-stationarity and quasi-ergodicity of general Markov *** show,among other things,that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t,x,y)(with respect to m)which is bounded in(x,y)for every t>0,then X has a unique quasi-stationary distribution and a unique quasi-ergodic *** also present several classes of Markov processes satisfying the above conditions.

关键词： Markov processes quasi-stationary distributions mean ratio quasi-stationary distributions quasiergodicity distributions

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分枝Markov过程的脊柱分解及其应用

中国科学：数学 2021年第11期51卷 1819-1844页

作者：任艳霞宋仁明北京大学数学科学学院北京100871 Department of Mathematics University of IllinoisUrbanaIL 61801USA 四川大学数学学院成都610064

现有分枝Markov过程脊柱分解的构造都有一个假设条件:单个粒子的后代个数>1.本文给出一般分枝Markov过程脊柱分解的详细构造,并允许单个粒子后代个数为0.然后,给出脊柱分解的一些应用.

关键词：分枝Brown运动分枝Hunt过程 Kesten-Stigum 定理脊柱分解定理行波解

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分枝马氏过程与超过程的中心极限定理

河北大学学报（自然科学版） 2021年第5期41卷 463-480页

作者：任艳霞宋仁明北京大学数学科学学院北京100871 伊利诺伊大学数学系厄巴纳G1801 四川大学数学系四川成都610065

给出近几年来在分枝马氏过程与超过程的极限理论方面得到的部分结果.主要包括分枝机制满足二阶矩条件的分枝Hunt过程的中心极限定理;分枝机制满足二阶矩条件的一般超过程的中心极限定理;分枝机制是空间齐次的稳定分枝机制的超OU过程的... 详细信息

给出近几年来在分枝马氏过程与超过程的极限理论方面得到的部分结果.主要包括分枝机制满足二阶矩条件的分枝Hunt过程的中心极限定理;分枝机制满足二阶矩条件的一般超过程的中心极限定理;分枝机制是空间齐次的稳定分枝机制的超OU过程的中心极限定理.文章同时涉及到大数律结果.

关键词：分枝Hunt过程超过程大数律中心极限定理

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上临界超过程的一类强极限的性质

中国科学：数学 2019年第3期49卷 485-504页

作者：任艳霞宋仁明张蕊北京大学数学科学学院北京100871 Department of Mathematics University of Illinois 南开大学数学科学学院天津300071 首都师范大学数学科学学院北京100048

假设X={X_t, t≥0; P_μ}是局部紧可分距离空间E上的上临界超过程,Ф_0是X的均值半群的生成元的与第一特征值λ_0对应的正特征函数,则M_t:=e^(-λ_0t)是非负鞅.令M_∞是M_t的极限,则M_∞是非退化的当且仅当L log L条件成立.当L log L条... 详细信息

假设X={X_t, t≥0; P_μ}是局部紧可分距离空间E上的上临界超过程,Ф_0是X的均值半群的生成元的与第一特征值λ_0对应的正特征函数,则M_t:=e^(-λ_0t)是非负鞅.令M_∞是M_t的极限,则M_∞是非退化的当且仅当L log L条件成立.当L log L条件不一定成立时,最近, Ren等(2017)证明了存在定义在[0,∞)上的非负函数γ_t及非退化随机变量W使得对任意E上非零Borel有限测度μ,有lim_(t→∞)

关键词：超过程上临界非退化强极限绝对连续小值概率尾概率