We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the nat...We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn｜ξ], the convergence rates of W - Wn （by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability）, and limit theorems （such as central limit theorems, moderate and large deviations principles） on （log Zn）.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
This paper is a continuation of our recent paper(Electron.J.Probab.,24(141),(2019))and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes(X_(t))t≥0 with branching mec...This paper is a continuation of our recent paper(Electron.J.Probab.,24(141),(2019))and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes(X_(t))t≥0 with branching mechanisms of infinite second moments.In the aforementioned paper,we proved stable central limit theorems for X_(t)(f)for some functions f of polynomial growth in three different regimes.However,we were not able to prove central limit theorems for X_(t)(f)for all functions f of polynomial growth.In this note,we show that the limiting stable random variables in the three different regimes are independent,and as a consequence,we get stable central limit theorems for X_(t)(f)for all functions f of polynomial growth.展开更多
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.
The potential energy surface of O（^1D）＋C2H5Cl reaction was studied using QCISD（T）/6- 311＋＋G（d,p）//MP2/6-31G（d,p） method. The calculations reveal an insertion-elimination mechanism. The insertion reaction o...The potential energy surface of O（^1D）＋C2H5Cl reaction was studied using QCISD（T）/6- 311＋＋G（d,p）//MP2/6-31G（d,p） method. The calculations reveal an insertion-elimination mechanism. The insertion reaction of O（^1D） and C2H5Cl produces two energy-rich intermediates, IM1 and IM2, which subsequently decompose into various products. The calculations of the branching ratios of various products formed through the two intermediates were carried out using RRKM （Rice-Ramsperger-Kassel-Marcus） theory at the collision energies of 0, 20.9, 41.8, 62.7, 83.6, 104.5, and 125.4 kJ/mol. HCl is the main decomposition product for IM1; CH2OH is the main decomposition product for IM2. Since IM1 is more stable than IM2, HCl is probably the main product of the O（^1D）＋C2H5Cl reaction.展开更多
First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional ...First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification ofβ-MBCRE according to the different standards.展开更多
基金Acknowledgements This work was partially supported by the National Natural Science Foundation of China （Grant Nos. 11171044, 11101039） and the Natural Science Foundation of Hunan Province （Grant No. 11JJ2001）.
文摘We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn｜ξ], the convergence rates of W - Wn （by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability）, and limit theorems （such as central limit theorems, moderate and large deviations principles） on （log Zn）.
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supported in part by NSFC(Grant Nos.11731009 and 12071011)the National Key R&D Program of China(Grant No.2020YFA0712900)supported in part by Simons Foundation(#429343,Renming Song)。
文摘This paper is a continuation of our recent paper(Electron.J.Probab.,24(141),(2019))and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes(X_(t))t≥0 with branching mechanisms of infinite second moments.In the aforementioned paper,we proved stable central limit theorems for X_(t)(f)for some functions f of polynomial growth in three different regimes.However,we were not able to prove central limit theorems for X_(t)(f)for all functions f of polynomial growth.In this note,we show that the limiting stable random variables in the three different regimes are independent,and as a consequence,we get stable central limit theorems for X_(t)(f)for all functions f of polynomial growth.
文摘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.
基金ACKNOWLEDGMENT This work was supported Science Foundation of China by the National Natural （No.50772107）.
文摘The potential energy surface of O（^1D）＋C2H5Cl reaction was studied using QCISD（T）/6- 311＋＋G（d,p）//MP2/6-31G（d,p） method. The calculations reveal an insertion-elimination mechanism. The insertion reaction of O（^1D） and C2H5Cl produces two energy-rich intermediates, IM1 and IM2, which subsequently decompose into various products. The calculations of the branching ratios of various products formed through the two intermediates were carried out using RRKM （Rice-Ramsperger-Kassel-Marcus） theory at the collision energies of 0, 20.9, 41.8, 62.7, 83.6, 104.5, and 125.4 kJ/mol. HCl is the main decomposition product for IM1; CH2OH is the main decomposition product for IM2. Since IM1 is more stable than IM2, HCl is probably the main product of the O（^1D）＋C2H5Cl reaction.
基金supported by the National Natural Science Foundation of China(Grant No:10371092)the Foundation of Wuhan University.
文摘First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification ofβ-MBCRE according to the different standards.
