We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection.The existence criterion of the process is firstly considered.We prove that if the su...We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection.The existence criterion of the process is firstly considered.We prove that if the sum of the resurrection rates is finite,then there does not exist any process.An existence criterion is then established when the sum of the resurrection rates is infinite.Some equivalent criteria,possessing the advantage of being easily checked,are obtained for the latter case.The uniqueness criterion for such process is also investigated.We prove that although there exist infinitely many of them,there always exists a unique honest process for a given q-matrix.This unique honest process is then constructed.The ergodicity property of this honest process is analysed in detail.We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.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 = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process...We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) 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(ξn) on R+, and reproduce independently new particles according to a probability law p(ξn) on N. 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 EξZ(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
We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching mo...We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Z+2 \ 0. It is shown that this λC can be directly obtained from the generating functions of the corresponding q-matrix. Moreover, the λC-invariant measures/vectors and quasi-distributions of such processes are deeply considered. A λC-invariant vector for the q-matrix (or for the process) on C is given and the generating functions of λC-invariant measures and quasi-stationary distributions for the process on C are presented.展开更多
A new class of branching models,the general collision branching processes with two parameters,is considered in this paper.For such models,it is necessary to evaluate the absorbing probabilities and mean extinction tim...A new class of branching models,the general collision branching processes with two parameters,is considered in this paper.For such models,it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing states.Regularity and uniqueness criteria are firstly established.Explicit expressions are then obtained for the extinction probability vector,the mean extinction times and the conditional mean extinction times.The explosion behavior of these models is investigated and an explicit expression for mean explosion time is established.The mean global holding time is also obtained.It is revealed that these properties are substantially different between the super-explosive and sub-explosive cases.展开更多
We simply call a superprocess conditioned on non-extinction a conditioned superprocess. In this study, we investigate some properties of the conditioned superprocesses (subcritical or critical). Firstly, we give an eq...We simply call a superprocess conditioned on non-extinction a conditioned superprocess. In this study, we investigate some properties of the conditioned superprocesses (subcritical or critical). Firstly, we give an equivalent description of the probability of the event that the total occupation time measure on a compact set is finite and some applications of this equivalent description. Our results are extensions of those of Krone (1995) from particular branching mechanisms to general branching mechanisms. We also prove a claim of Krone for the cases of d = 3, 4. Secondly, we study the local extinction property of the conditioned binary super-Brownian motion {Xt, P μ∞ }. When d = 1, as t goes to infinity, Xt/√t converges to ηλ in weak sense under P μ∞ , where η is a nonnegative random variable and λ is the Lebesgue measure on R. When d 2, the conditioned binary super-Brownian motion is locally extinct under P μ∞ .展开更多
In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then stud...In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then study the large and moderate deviations associated with this central tendency.展开更多
We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting proc...We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting processes are Gaussian processes.展开更多
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).
基金partially supported by the National Natural Science Foundation of China (Grant No. 10771216)Research Grants Council of Hong Kong (Grant No. HKU 7010/06P)Project-sponsored by SRF for ROCS,SEM
文摘We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection.The existence criterion of the process is firstly considered.We prove that if the sum of the resurrection rates is finite,then there does not exist any process.An existence criterion is then established when the sum of the resurrection rates is infinite.Some equivalent criteria,possessing the advantage of being easily checked,are obtained for the latter case.The uniqueness criterion for such process is also investigated.We prove that although there exist infinitely many of them,there always exists a unique honest process for a given q-matrix.This unique honest process is then constructed.The ergodicity property of this honest process is analysed in detail.We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.
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 = (ξ0,ξ1,...) 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(ξn) on R+, and reproduce independently new particles according to a probability law p(ξn) on N. 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 EξZ(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. 10771216)Project sponsored by Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry(Grant No. [2007]1108)
文摘We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Z+2 \ 0. It is shown that this λC can be directly obtained from the generating functions of the corresponding q-matrix. Moreover, the λC-invariant measures/vectors and quasi-distributions of such processes are deeply considered. A λC-invariant vector for the q-matrix (or for the process) on C is given and the generating functions of λC-invariant measures and quasi-stationary distributions for the process on C are presented.
基金supported by National Natural Science Foundation of China (Grant No.10771216)Research Grants Council of Hong Kong (Grant No.HKU 7010/06P)Scientific Research Foundation for Returned Overseas Chinese Scholars,State Education Ministry of China (Grant No.[2007]1108)
文摘A new class of branching models,the general collision branching processes with two parameters,is considered in this paper.For such models,it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing states.Regularity and uniqueness criteria are firstly established.Explicit expressions are then obtained for the extinction probability vector,the mean extinction times and the conditional mean extinction times.The explosion behavior of these models is investigated and an explicit expression for mean explosion time is established.The mean global holding time is also obtained.It is revealed that these properties are substantially different between the super-explosive and sub-explosive cases.
基金supported by National Natural Science Foundation of China (Grant No. 10471003, 10871103)
文摘We simply call a superprocess conditioned on non-extinction a conditioned superprocess. In this study, we investigate some properties of the conditioned superprocesses (subcritical or critical). Firstly, we give an equivalent description of the probability of the event that the total occupation time measure on a compact set is finite and some applications of this equivalent description. Our results are extensions of those of Krone (1995) from particular branching mechanisms to general branching mechanisms. We also prove a claim of Krone for the cases of d = 3, 4. Secondly, we study the local extinction property of the conditioned binary super-Brownian motion {Xt, P μ∞ }. When d = 1, as t goes to infinity, Xt/√t converges to ηλ in weak sense under P μ∞ , where η is a nonnegative random variable and λ is the Lebesgue measure on R. When d 2, the conditioned binary super-Brownian motion is locally extinct under P μ∞ .
基金the National Natural Science Foundation of China (Grant No.10121101)
文摘In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then study the large and moderate deviations associated with this central tendency.
基金supported by National Natural Science Foundation of China (Grant No.10721091)
文摘We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting processes are Gaussian processes.
基金the Program for New Century Excellent Talents in University (Grant No. 05-0143)the National Natural Science Foundation of China (Grant No. 10721091)
文摘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).