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On the transience and recurrence of Lamperti’s random walk on Galton-Watson trees

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Science China Mathematics 2019年第9期62卷 1813-1822页

作者： wenming hong Minzhi Liu School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems Beijing Normal University

In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(x) =(λ +A/|x|α)-|x|, where |x| is the generation of the vertex x. For(Xn)n≥0, a C(x)-biased random walk on the tree, we show that (1) when λ≠ m, α > 0,(Xn)n≥0 is transient/recurrent according to whether λ m, respectively;(2) when λ = m, 0 0, *** particular, if P(N = 1) = 1, the C(x)-biased random walk is Lamperti’s random walk on the nonnegative integers(see Lamperti(1960)).

关键词： inhomogeneous biased random walk on the branching tree transience recurrence time-inhomogeneous branching process

Scaling limit theorem for transient random walk in random environment

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Frontiers of Mathematics in China 2018年第5期13卷 1033-1044页

作者： wenming hong Hui YANG School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems Beijing Normal University Beijing 100875 China College of Science Minzu University of China Beijing 100081 China

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of co... 详细信息

关键词： Random walk random environment diffusion process Brownian motion with drift

Extremum of a time-inhomogeneous branching random walk

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Frontiers of Mathematics in China 2021年第2期16卷 459-478页

作者： Wanting HOU Xiaoyue ZHANG wenming hong Department of Mathematics Northeastern UniversityShenyang 110004China School of Statistics Capital University of Economics and BusinessBeijing 100070China School of Mathematical Sciences&Laboratory of Mathematics and Complex Systems Beijing Normal UniversityBeijing 100875China

Consider a time-inhomogeneous branching random walk, generated by the point process Ln which composed by two independent parts: ‘branching’offspring Xn with the mean 1+B(1+n)−β for β∈(0,1) and ‘displacement’ ξn with a drift A(1+n)^(−2α) for α∈(0,1/2), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ ξn is strictly positive or negative for |A|>0 but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter β and α.

关键词： Branching random walk time-inhomogeneous branching process random walk

Scaling limit of local time of Sinai＇s random walk

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Frontiers of Mathematics in China 2015年第6期10卷 1313-1324页

作者： wenming hong Hui YANG Ke ZHOU School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems Beijing Normal University Beijing 100875 China School of Statistics University of International Business and Economics Beijing 100029China

We prove that the local times of a sequence of Sinai＇s random walks converge to those of Brox＇s diffusion by proper scaling. Our proof is based on the intrinsic branching structure of the random walk and the converg... 详细信息

关键词： Sinai's random walk random environment local time Brox's diffusion branching process

Markov Processes and Related Topics(Ⅲ)

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Frontiers of Mathematics in China 2021年第2期16卷 255-256页

作者： wenming hong Yonghua MAO Yuhui ZHANG Research Center for Stochastics School of Mathematical SciencesBeijing Normal UniversityBeijing 100875China

It is our great pleasure to present the third special issue for the Markov Processes and Related Topics,on the occasion of the 20th anniversary of the Research Center for Stochastics of Beijing Normal University(BNU),supported by the innovative research group project of the National Natural Science Foundation of *** 2002,the Research Center for Stochastics started toorganize'Workshop on Markov Processes and Related Topics'.During the past 20 years,the workshop held almost each year except some special situations(for example,2003 SARS and 2020 COVID-19).Among the 16 workshops,11 of them were or will be co-organized with some universities out of Beijing。

关键词： China. Markov workshop

Light-tailed behavior of stationary distribution for state-dependent random walks on a strip

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Frontiers of Mathematics in China 2014年第4期9卷 813-834页

作者： wenming hong Meijuan ZHANG Yiqiang Q. ZHAO School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems Beijing Normal University Beijing 100875 China School of Statistics and Mathematics Central University of Finance and EconomicsBeijing 100081 China School of Mathematics and Statistics Carleton University Ottawa Ont K1S 5B6 Canada

We consider the state-dependent reflecting random walk on a half- strip. We provide explicit criteria for （positive） recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method.

关键词： Random walk on a strip stationary distribution light-tailedbehavior branching process recurrence state-dependent

THE OCCUPATION DENSITY FIELD FOR THE CATALYTIC SUPER-BROWNIAN MOTION

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Chinese Annals of Mathematics,Series B 1999年第4期20卷 447-454页

作者： hong wenming(Institute of Mathematics, Fudan University Shanghai 200433, China) Fudan Univ Inst Math Shanghai 200433 Peoples R China

It is proved that the occupation time of the catalytic super-Brownian motion is absolutely continuous for d = 1, and the occupation density field is jointly continuous and jointly Holder continuous.

关键词： Branching rate functional Brownian collision local time Catalytic super-Brownian motion Occupation time Occupation density field

RENEWAL THEOREM FOR(L,1)-RANDOM WALK IN RANDOM ENVIRONMENT

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Acta Mathematica Scientia 2013年第6期33卷 1736-1748页

作者：洪文明孙鸿雁 Laboratory of Mathematics and Complex Systems School of Mathematical SciencesBeijing Normal University

We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in hong and Wang the intrinsic branching structure within the （2013）.

关键词： random walk in random environment renewal theorem multitype branchingprocess in random environment coupling

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration

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Science China Mathematics 2008年第3期51卷 343-350页

作者： hong wenming School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education,Beijing Normal University, Beijing 100875, China School of Mathematical Sciences Laboratory of Mathematics Complex Systems Ministry of Education Beijing Normal University Beijing 100875 China

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3≤d≤6, which fills in the gap between central limit theorem(CLT)and large deviation principle(LDP).

关键词： super-Brownian motion large deviation moderate deviation random immigration Primary 60J80 Secondary 60F05