In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTR...In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors es-tablish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov br...展开更多
We consider a branching random walk in an independent and identically distributed random environment ξ=(ξn) indexed by the time. Let W be the limit of the martingale Wn=∫e^-txZn(dx)/Eξ∫e^-txZn(dx), with Zn denoti...We consider a branching random walk in an independent and identically distributed random environment ξ=(ξn) indexed by the time. Let W be the limit of the martingale Wn=∫e^-txZn(dx)/Eξ∫e^-txZn(dx), with Zn denoting the counting measure of particles of generation n, and Eξ the conditional expectation given the environment ξ. We find necessary and sufficient conditions for the existence of quenched moments and weighted moments of W, when W is non-degenerate.展开更多
We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environmen...We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The envi- ronment is supposed to be independent and identically distributed. For A C R, let Zn（A） be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn （-） with appropriate normalization.展开更多
Let(Z_(n))be a supercritical bisexual branching process in a random environmentξ.We study the almost sure(a.s.)convergence rate of the submartingale W_(n)=Z_(n)/In to its limit W,where(In)is an usually used norming s...Let(Z_(n))be a supercritical bisexual branching process in a random environmentξ.We study the almost sure(a.s.)convergence rate of the submartingale W_(n)=Z_(n)/In to its limit W,where(In)is an usually used norming sequence.We prove that under a moment condition of order p∈(1,2),W-W_(n)=o(e^(-na))a.s.for some a>0 that we find explicitly;assuming the logarithmic moment condition holds,we haveW-W_(n)=o(n^(-α))a.s..In order to obtain these results,we provide the L^(p)-convergence of(W_(n));similar conclusions hold for a bisexual branching process in a varying environment.展开更多
We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the converge...We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].展开更多
We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on ? with a random env...We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on ? with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.展开更多
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the proce...We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.展开更多
Let W be the limit of the normalized population size of a supercritical branching process in a varying or random environment. By an elementary method, we find sufficient conditions under which W has finite weighted mo...Let W be the limit of the normalized population size of a supercritical branching process in a varying or random environment. By an elementary method, we find sufficient conditions under which W has finite weighted moments of the form EWpl(W), where p > 1, l 0 is a concave or slowly varying function.展开更多
In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of ｜W｜ is of order t-κ as t→∞, ...In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of ｜W｜ is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for （L-1） random walk in random environment, generalizing the stable law for the nearest random walk in random environment （see ＂Kesten, Kozlov, Spitzer： A limit law for random walk in a random environment. Compositio Math., 30, 145-168 （1975）＂）.展开更多
We consider anR^(d)-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_(n)(z)(z∈C^(d))be the natural complex martingale of the process.We s...We consider anR^(d)-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_(n)(z)(z∈C^(d))be the natural complex martingale of the process.We show necessary and sufficient conditions for the L^(α)-convergence of W_(n)(z)forα>1,as well as its uniform convergence region.展开更多
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...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 convergence of the branching processes in random environment.展开更多
基金Supported by the National Natural Science Foundation of China (10771185 and 10871200)
文摘In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors es-tablish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov br...
基金benefited from the support of the French government Investissements d’Avenir program ANR-11-LABX-0020-01partially supported by the National Natural Science Foundation of China(11571052,11401590,11731012 and 11671404)by Hunan Natural Science Foundation(2017JJ2271)
文摘We consider a branching random walk in an independent and identically distributed random environment ξ=(ξn) indexed by the time. Let W be the limit of the martingale Wn=∫e^-txZn(dx)/Eξ∫e^-txZn(dx), with Zn denoting the counting measure of particles of generation n, and Eξ the conditional expectation given the environment ξ. We find necessary and sufficient conditions for the existence of quenched moments and weighted moments of W, when W is non-degenerate.
基金partially supported by the National Natural Science Foundation of China(NSFC,11101039,11171044,11271045)a cooperation program between NSFC and CNRS of France(11311130103)+1 种基金the Fundamental Research Funds for the Central UniversitiesHunan Provincial Natural Science Foundation of China(11JJ2001)
文摘We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The envi- ronment is supposed to be independent and identically distributed. For A C R, let Zn（A） be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn （-） with appropriate normalization.
基金supported by the Fundamental Research Funds for the Central University (Grant No.19JNLH09)Innovation Team Project in Guangdong Province,P.R.China (Grant No.2016WCXTD004)+1 种基金supported by the National Natural Science Foundation of China (Grants no.11731012,12271062)Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science&Technology)。
文摘Let(Z_(n))be a supercritical bisexual branching process in a random environmentξ.We study the almost sure(a.s.)convergence rate of the submartingale W_(n)=Z_(n)/In to its limit W,where(In)is an usually used norming sequence.We prove that under a moment condition of order p∈(1,2),W-W_(n)=o(e^(-na))a.s.for some a>0 that we find explicitly;assuming the logarithmic moment condition holds,we haveW-W_(n)=o(n^(-α))a.s..In order to obtain these results,we provide the L^(p)-convergence of(W_(n));similar conclusions hold for a bisexual branching process in a varying environment.
基金Acknowledgements The authors would like to thank the anonymous referees for valuable comments and remarks. This work was partially supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology （HIT. NSRIF. 2015102）, the National Natural Science Foundation of China （Grant Nos. 11171044, 11101039）, and by the Natural Science Foundation of Hunan Province （Grant No. 11JJ2001）.
文摘We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].
基金the National Natural Science Foundation of China (Grant Nos. 10271020,10471012)SRF for ROCS, SEM (Grant No. [2005]564)
文摘We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on ? with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.
基金the National Natural Sciente Foundation of China (Grant Nos. 10771021, 10471012)Scientific Research Foundation for Returned Scholars, Ministry of Education of China (Grant No. [2005]564)
文摘We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.
基金supported by National Natural Science Foundation of China (Grant No. 10771021)Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20104306110001)+1 种基金the Planned Science and Technology Project of Hunan Province (Grant Nos. 2010fj6036, 2009fi3098)the Scientific Research Fund of Hunan Provincial Education Department (Grant Nos. 08C120, 09C113, 09C059)
文摘Let W be the limit of the normalized population size of a supercritical branching process in a varying or random environment. By an elementary method, we find sufficient conditions under which W has finite weighted moments of the form EWpl(W), where p > 1, l 0 is a concave or slowly varying function.
基金Supported by National Nature Science Foundation of China(Grant No.11226199)
文摘In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of ｜W｜ is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for （L-1） random walk in random environment, generalizing the stable law for the nearest random walk in random environment （see ＂Kesten, Kozlov, Spitzer： A limit law for random walk in a random environment. Compositio Math., 30, 145-168 （1975）＂）.
基金This work was supported in part by the National Natural Science Foundation of China(Nos.11601019,11971063,11501146)the Scientific Research Project of Beijing Municipal Education(Grant No.SQKM201610011006).
文摘We consider anR^(d)-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_(n)(z)(z∈C^(d))be the natural complex martingale of the process.We show necessary and sufficient conditions for the L^(α)-convergence of W_(n)(z)forα>1,as well as its uniform convergence region.
基金Supported by NSFC of China(11171101,11171044)Key Laboratory of High Performance Computing and Stochastic Information Processing(HPCSIP,Hunan Normal University)(Ministry of Education of China)+4 种基金Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities(12FEFM06)Hunan Provincial Natural Science Foundation of China(11JJ2001,13JJ6048)Research Fund for Doctoral Program of Higher Education of China(20104306110001)Planned Science and Technology Project of Hunan Province(2010fj 6036,2009fi3098)Scientific Research Fund of Hunan Provincial Education Department(12C0027,08C120,09C113,09C059)
基金Acknowledgements The authors thank the anonymous referees for their very careful reading of the manuscript. This work was supported in part by the National Natural Science Foundation of China （Grant No. 10721091） and the 985-project.
文摘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 convergence of the branching processes in random environment.
