In this paper, we study the logarithmic Sobolev inequalities for two-sided birth-death processes. An estimate of the logarithmic Sobolev constant α for a two-sided birth-death process is obtained by the Hardy-type in...In this paper, we study the logarithmic Sobolev inequalities for two-sided birth-death processes. An estimate of the logarithmic Sobolev constant α for a two-sided birth-death process is obtained by the Hardy-type inequality and a criteria for a is also presented.展开更多
In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and driv...In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and drive the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.展开更多
The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized tha...The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event.展开更多
A new structure with the special property that instantaneous state and catas-trophes is imposed to ordinary birth-death processes is considered. Kendall's conjecture forthe processes is proved to be right.
By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2 × 2 games in a well-mixed population with a finite size is investigated. In the model, the individuals＇ payoff accumu...By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2 × 2 games in a well-mixed population with a finite size is investigated. In the model, the individuals＇ payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when s is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the ＇Hawk-Dove＇ game and the ＇Coordinate＇ game are used to illustrate our model. We give the evolutionary stable strategy （ESS） of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.展开更多
Human Immunodeficiency Virus (HIV) dynamics in Africa are purely characterised by sparse sampling of DNA sequences for individuals who are infected. There are some sub-groups that are more at risk than the general pop...Human Immunodeficiency Virus (HIV) dynamics in Africa are purely characterised by sparse sampling of DNA sequences for individuals who are infected. There are some sub-groups that are more at risk than the general population. These sub-groups have higher infectivity rates. We came up with a likelihood inference model of multi-type birth-death process that can be used to make inference for HIV epidemic in an African setting. We employ a likelihood inference that incorporates a probability of removal from infectious pool in the model. We have simulated trees and made parameter inference on the simulated trees as well as investigating whether the model distinguishes between heterogeneous and homogeneous dynamics. The model makes fairly good parameter inference. It distinguishes between heterogeneous and homogeneous dynamics well. Parameter estimation was also performed under sparse sampling scenario. We investigated whether trees obtained from a structured population are more balanced than those from a non-structured host population using tree statistics that measure tree balance and imbalance. Trees from non-structured population were more balanced basing on Colless and Sackin indices.展开更多
In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the in...In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the inequalities, the variational formulas andexplicit bounds of the corresponding constants in the inequalities are presented. As typicalapplications, the Nash inequalities and logarithmic Sobolev inequalities are examined.展开更多
In this paper, we consider birth-death processes on a tree T and we are interested when it is regular, recurrent and ergodic (strongly, exponentially). By constructing two corresponding birth death processes on Z+, we...In this paper, we consider birth-death processes on a tree T and we are interested when it is regular, recurrent and ergodic (strongly, exponentially). By constructing two corresponding birth death processes on Z+, we obtain computable conditions sufficient or necessary for that (in many cases, these two conditions coincide). With the help of these constructions, we give explicit upper and lower bounds for the Dirichlet eigenvalue λ0. At last, some examples are investigated to justify our results.展开更多
We restudy the master-equation approach to aggregation in freeway traffic based on the theory of birth-death process, in which the clustering behaviour in one-lane freeway traffic model is investigated. The transition...We restudy the master-equation approach to aggregation in freeway traffic based on the theory of birth-death process, in which the clustering behaviour in one-lane freeway traffic model is investigated. The transition probabilities for the jump processes are reconstructed by using Greenshields＇ model, and the equation of the mean size of the cluster at any time t is derived from the birth^death equation. Numerical experiments show the clustering behaviours varying with time very well.展开更多
In converged heterogeneous wireless networks, vertical handoff is an important issue in radio resource management and occurs when an end user switches from one network to another （e.g., from wireless local area netwo...In converged heterogeneous wireless networks, vertical handoff is an important issue in radio resource management and occurs when an end user switches from one network to another （e.g., from wireless local area network to wideband code division multiple access）. Efficient vertical handoff should allocate network resources efficiently and maintain good quality of service （QoS） for the end users. The objective of this work is to determine conditions under which vertical handoff can be performed. The channel usage situation of each access network is formulated as a birth-death process with the objective of predicting the avaliable bandwidth and the blocking probability. A reward function is used to capture the network bandwidth and the blocking probability is expressed as a cost function. An end user will access the certain network which maximizes the total function defined as the combination of the reward fimction and the cost function. Simulation results show that the proposed algorithm can significantly improve the network performance, including higher bandwidth for end users and lower new call blocking and handoff call blocking probability for networks.展开更多
Massive multiple-input multiple-output(MIMO)emerges as one of the most promising technologies for 5G mobile communication systems.Compared to the conventional MIMO channel models,channel researches and measurements sh...Massive multiple-input multiple-output(MIMO)emerges as one of the most promising technologies for 5G mobile communication systems.Compared to the conventional MIMO channel models,channel researches and measurements show that significant nonstationary properties rise in massive MIMO channels.Therefore,an accurate channel model is indispensable for the sake of massive MIMO system design and performance evaluation.This article presents an overview of methods of modeling non-stationary properties on both the array and time axes,which are mainly divided into two major categories:birth-death(BD)process and cluster visibility region(VR)method.The main concepts and theories are described,together with useful implementation guidelines.In conclusion,a comparison between these two methods is made.展开更多
In this paper, we present an explicit and computable lower bound for the first eigenvalue of birth-death processes with killing. This estimate is qualitatively sharp for birth-death processes without killing. We also ...In this paper, we present an explicit and computable lower bound for the first eigenvalue of birth-death processes with killing. This estimate is qualitatively sharp for birth-death processes without killing. We also establish an approximation procedure for the first eigenvalue of the birth-death process with killing by an increasing principal eigenvalue sequence of some birth-death processes without killing. Some applications of our results are illustrated by many examples.展开更多
This paper is a continuation of the study of the algebraic speed for Markov processes. The authors concentrate on algebraic decay rate for the transient birth-death processes. According to the classification of the bo...This paper is a continuation of the study of the algebraic speed for Markov processes. The authors concentrate on algebraic decay rate for the transient birth-death processes. According to the classification of the boundaries, a series of the sufficient conditions for algebraic decay is presented. To illustrate the power of the results, some examples are included.展开更多
Constructing some proper functional spaces, we obtain the corresponding norm for the operator （-.L）^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Fr...Constructing some proper functional spaces, we obtain the corresponding norm for the operator （-.L）^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Front. Math. China, 2010, 5（3）： 379-515], for transient birth-death processes.展开更多
We illustrate the influence of an external periodic force and noise on a physical system by the example of an oscillator. These two forces seem to be the reverse of each other, since the latter leads to disorder while...We illustrate the influence of an external periodic force and noise on a physical system by the example of an oscillator. These two forces seem to be the reverse of each other, since the latter leads to disorder while the former works in an orderly fashion. Nevertheless, it is shown that they may influence a system in a similar way, sometime even substituting for one another. These examples serve to illustrate one of the main achievements of twentieth-century physics, which has established that deterministic and random phenomena complement rather than contradict each other.展开更多
In this paper. we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product. we prove that for these processes. transience implies that the semigroups P(t) ...In this paper. we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product. we prove that for these processes. transience implies that the semigroups P(t) decay as ‖P(t)‖_(1--x)≤Ct^(-1). Sufficient conditions for general Msrkov chains are also obtained.展开更多
By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in ...By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.展开更多
For a reversible quasi-birth and death process, we generalize and refine the decomposition method, by constructing a birth-death process and a sequence of restriction processes. The spectral gap for the quasi-birth an...For a reversible quasi-birth and death process, we generalize and refine the decomposition method, by constructing a birth-death process and a sequence of restriction processes. The spectral gap for the quasi-birth and death process is estimated in terms of the spectral gaps for these processes, and in some special cases, the estimation is sharp. With the aid of the symmetrization procedure, the result is also applied to two queueing models： M/M/1 in random environment and MIMIc with synchronous vacation.展开更多
Suppose {X（t）; t≥ 0} is a single birth process with birth rate qii＋l （i 〉 0） and death rate qij （i 〉 j ≥ 0）. It is proved in this paper that （i） if there exists aconstant c≥ 0 such that b（i）-a（i）＋ci...Suppose {X（t）; t≥ 0} is a single birth process with birth rate qii＋l （i 〉 0） and death rate qij （i 〉 j ≥ 0）. It is proved in this paper that （i） if there exists aconstant c≥ 0 such that b（i）-a（i）＋ci is nondecreasing with respect to i and a（i） ＋ u（i） - ci ≥ 0 （i≥ 0）, then VarX（t）-EX（t）≥-X（0）e^-2ct,t≥0,or （ii） if there exists a constant u（i） - c≥ 0 such that b（i）-a（i）＋ci is non-increasing with respect to i and a（i）＋u（i）-ci≤0（i≥0）,then VarX（t） - EX（t） ≤ -X（0）e^-2c,t ≥ 0 Hereb（i） = qii＋1, a（0） = 0, a（i） = ∑j=^ijqii-j （i≥ 1）, u（0） = u（1） =0 and u（i） = 1/2∑j=^ij（j - 1）qii-j （i ≥ 2） . This result covers the results for birth-death processes obtained in [7].展开更多
In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusio...In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.展开更多
基金the National Natural Science Foundation of China(10271091)
文摘In this paper, we study the logarithmic Sobolev inequalities for two-sided birth-death processes. An estimate of the logarithmic Sobolev constant α for a two-sided birth-death process is obtained by the Hardy-type inequality and a criteria for a is also presented.
文摘In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and drive the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.
基金supported in part by the NIH grant R01CA241134supported in part by the NSF grant CMMI-1552764+3 种基金supported in part by the NSF grants DMS-1349724 and DMS-2052465supported in part by the NSF grant CCF-1740761supported in part by the U.S.-Norway Fulbright Foundation and the Research Council of Norway R&D Grant 309273supported in part by the Norwegian Centennial Chair grant and the Doctoral Dissertation Fellowship from the University of Minnesota.
文摘The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event.
基金Supported by the Guangxi Science Foundation(0339071)
文摘A new structure with the special property that instantaneous state and catas-trophes is imposed to ordinary birth-death processes is considered. Kendall's conjecture forthe processes is proved to be right.
基金supported by the National Natural Science Foundation of China (Grant No. 71071119)the Fundamental Research Funds for the Central Universities
文摘By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2 × 2 games in a well-mixed population with a finite size is investigated. In the model, the individuals＇ payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when s is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the ＇Hawk-Dove＇ game and the ＇Coordinate＇ game are used to illustrate our model. We give the evolutionary stable strategy （ESS） of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.
文摘Human Immunodeficiency Virus (HIV) dynamics in Africa are purely characterised by sparse sampling of DNA sequences for individuals who are infected. There are some sub-groups that are more at risk than the general population. These sub-groups have higher infectivity rates. We came up with a likelihood inference model of multi-type birth-death process that can be used to make inference for HIV epidemic in an African setting. We employ a likelihood inference that incorporates a probability of removal from infectious pool in the model. We have simulated trees and made parameter inference on the simulated trees as well as investigating whether the model distinguishes between heterogeneous and homogeneous dynamics. The model makes fairly good parameter inference. It distinguishes between heterogeneous and homogeneous dynamics well. Parameter estimation was also performed under sparse sampling scenario. We investigated whether trees obtained from a structured population are more balanced than those from a non-structured host population using tree statistics that measure tree balance and imbalance. Trees from non-structured population were more balanced basing on Colless and Sackin indices.
文摘In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the inequalities, the variational formulas andexplicit bounds of the corresponding constants in the inequalities are presented. As typicalapplications, the Nash inequalities and logarithmic Sobolev inequalities are examined.
基金supported by National Natural Science Foundation of China (Grant No.10721091)
文摘In this paper, we consider birth-death processes on a tree T and we are interested when it is regular, recurrent and ergodic (strongly, exponentially). By constructing two corresponding birth death processes on Z+, we obtain computable conditions sufficient or necessary for that (in many cases, these two conditions coincide). With the help of these constructions, we give explicit upper and lower bounds for the Dirichlet eigenvalue λ0. At last, some examples are investigated to justify our results.
基金Project supported by the National Natural Science Foundation of China （Grant No 10435080）, the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, China.
文摘We restudy the master-equation approach to aggregation in freeway traffic based on the theory of birth-death process, in which the clustering behaviour in one-lane freeway traffic model is investigated. The transition probabilities for the jump processes are reconstructed by using Greenshields＇ model, and the equation of the mean size of the cluster at any time t is derived from the birth^death equation. Numerical experiments show the clustering behaviours varying with time very well.
基金Project(20040533035) supported by the National Research Foundation for the Doctoral Program of Higher Education of ChinaProject (50275150) supported by the National Natural Science Foundation of China
文摘In converged heterogeneous wireless networks, vertical handoff is an important issue in radio resource management and occurs when an end user switches from one network to another （e.g., from wireless local area network to wideband code division multiple access）. Efficient vertical handoff should allocate network resources efficiently and maintain good quality of service （QoS） for the end users. The objective of this work is to determine conditions under which vertical handoff can be performed. The channel usage situation of each access network is formulated as a birth-death process with the objective of predicting the avaliable bandwidth and the blocking probability. A reward function is used to capture the network bandwidth and the blocking probability is expressed as a cost function. An end user will access the certain network which maximizes the total function defined as the combination of the reward fimction and the cost function. Simulation results show that the proposed algorithm can significantly improve the network performance, including higher bandwidth for end users and lower new call blocking and handoff call blocking probability for networks.
基金supported in part by the National Natural Science of Foundation for Creative Research Groups of China under Grant No.61421061Huawei Innovation Research Program.
文摘Massive multiple-input multiple-output(MIMO)emerges as one of the most promising technologies for 5G mobile communication systems.Compared to the conventional MIMO channel models,channel researches and measurements show that significant nonstationary properties rise in massive MIMO channels.Therefore,an accurate channel model is indispensable for the sake of massive MIMO system design and performance evaluation.This article presents an overview of methods of modeling non-stationary properties on both the array and time axes,which are mainly divided into two major categories:birth-death(BD)process and cluster visibility region(VR)method.The main concepts and theories are described,together with useful implementation guidelines.In conclusion,a comparison between these two methods is made.
文摘In this paper, we present an explicit and computable lower bound for the first eigenvalue of birth-death processes with killing. This estimate is qualitatively sharp for birth-death processes without killing. We also establish an approximation procedure for the first eigenvalue of the birth-death process with killing by an increasing principal eigenvalue sequence of some birth-death processes without killing. Some applications of our results are illustrated by many examples.
基金supported by the National Natural Science Foundation of China (No. 10721091)
文摘This paper is a continuation of the study of the algebraic speed for Markov processes. The authors concentrate on algebraic decay rate for the transient birth-death processes. According to the classification of the boundaries, a series of the sufficient conditions for algebraic decay is presented. To illustrate the power of the results, some examples are included.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11101040, 11131003）, the 985 Project, the 973 Project （No. 2011CB808000）, the Specialized Research Fund for the Doctoral Program of Higher Education （No. 20100003110005）, and the Fundamental Research Funds for the Central Universities.
文摘Constructing some proper functional spaces, we obtain the corresponding norm for the operator （-.L）^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Front. Math. China, 2010, 5（3）： 379-515], for transient birth-death processes.
文摘We illustrate the influence of an external periodic force and noise on a physical system by the example of an oscillator. These two forces seem to be the reverse of each other, since the latter leads to disorder while the former works in an orderly fashion. Nevertheless, it is shown that they may influence a system in a similar way, sometime even substituting for one another. These examples serve to illustrate one of the main achievements of twentieth-century physics, which has established that deterministic and random phenomena complement rather than contradict each other.
基金Research supported in part by RFDP (No 96002704)NSFC (No 19771008) Fok Ying-Tung Youth Foundation
文摘In this paper. we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product. we prove that for these processes. transience implies that the semigroups P(t) decay as ‖P(t)‖_(1--x)≤Ct^(-1). Sufficient conditions for general Msrkov chains are also obtained.
基金Supported by Foundation of Fujian’s Ministry of Education (Grant Nos. JA10058 and JA11051)National Natural Science Foundation of China (Grant No. 11126350)
文摘By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.
基金Supported in part by Program for New Century Excellent Talents in University (NCET)973 Project (Grant No. 2011CB808000)NSFC (Grant No. 10721091)
文摘For a reversible quasi-birth and death process, we generalize and refine the decomposition method, by constructing a birth-death process and a sequence of restriction processes. The spectral gap for the quasi-birth and death process is estimated in terms of the spectral gaps for these processes, and in some special cases, the estimation is sharp. With the aid of the symmetrization procedure, the result is also applied to two queueing models： M/M/1 in random environment and MIMIc with synchronous vacation.
基金Supported by the National Natural Science Foundation of China(No.10471130,10371024)
文摘Suppose {X（t）; t≥ 0} is a single birth process with birth rate qii＋l （i 〉 0） and death rate qij （i 〉 j ≥ 0）. It is proved in this paper that （i） if there exists aconstant c≥ 0 such that b（i）-a（i）＋ci is nondecreasing with respect to i and a（i） ＋ u（i） - ci ≥ 0 （i≥ 0）, then VarX（t）-EX（t）≥-X（0）e^-2ct,t≥0,or （ii） if there exists a constant u（i） - c≥ 0 such that b（i）-a（i）＋ci is non-increasing with respect to i and a（i）＋u（i）-ci≤0（i≥0）,then VarX（t） - EX（t） ≤ -X（0）e^-2c,t ≥ 0 Hereb（i） = qii＋1, a（0） = 0, a（i） = ∑j=^ijqii-j （i≥ 1）, u（0） = u（1） =0 and u（i） = 1/2∑j=^ij（j - 1）qii-j （i ≥ 2） . This result covers the results for birth-death processes obtained in [7].
基金Supported in part by NSFC(Grant No.11771047)Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai(Grant No.2019RS1057)。
文摘In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.
