In this work,for a one-dimensional regime-switching diffusion process,we show that when it is positive recurrent,then there exists a stationary distribution,and when it is null recurrent,then there exists an invariant...In this work,for a one-dimensional regime-switching diffusion process,we show that when it is positive recurrent,then there exists a stationary distribution,and when it is null recurrent,then there exists an invariant measure. We also provide the explicit representation of the stationary distribution and invariant measure based on the hitting times of the process.展开更多
For a given probability density function p（x） on R^d, we construct a （non-stationary） diffusion process xt, starting at any point x in R^d, such that 1/T ∫_o^T δ（xt-x）dt converges to p（x） almost surely. The ...For a given probability density function p（x） on R^d, we construct a （non-stationary） diffusion process xt, starting at any point x in R^d, such that 1/T ∫_o^T δ（xt-x）dt converges to p（x） almost surely. The rate of this convergence is also investigated. To find this rate, we mainly use the Clark-Ocone formula from Malliavin calculus and the Girsanov transformation technique.展开更多
For an ergodic continuous-time Markov process with a particular state in its space,the authors provide the necessary and sufficient conditions for exponential and strong ergodicity in terms of the moments of the first...For an ergodic continuous-time Markov process with a particular state in its space,the authors provide the necessary and sufficient conditions for exponential and strong ergodicity in terms of the moments of the first hitting time on the state.An application to the queue length process of M/G/1 queue with multiple vacations is given.展开更多
In this paper, we prove some limit theorems for killed Brownian motion during its life time. The emphases are on quasi-stationarity and quasi-ergodicity and related problems. On one hand, using an eigenfunction expans...In this paper, we prove some limit theorems for killed Brownian motion during its life time. The emphases are on quasi-stationarity and quasi-ergodicity and related problems. On one hand, using an eigenfunction expansion for the transition density, we prove the existence and uniqueness of both quasi-stationary distribution (qsd) and mean ratio quasi-stationary distribution (mrqsd). The later is shown to be closely related to laws of large numbers (LLN) and to quasi-ergodicity. We further show that the mrqsd is the unique stationary distribution of a certain limiting ergodic diffusion process of the BM conditioned on not having been killed. We also show that a phase transition occurs from mrqsd to qsd. On the other hand, we study the large deviation behavior related to the above problems. A key observation is that the mrqsd is the unique minimum of certain large deviation rate function. We further prove that the limiting diffusion process also satisfies a large deviation principle with the rate function attaining its unique minimum at the mrqsd. These give interpretations of the mrqsd from different points of view, and establish some intrinsic connections among the above topics. Some general results concerning Yaglom limit, moment convergence and LLN are also obtained.展开更多
In this paper,we study the quasi-stationarity and quasi-ergodicity of general Markov processes.We show,among other things,that if X is a standard Markov process admitting a dual with respect to a finite measure m and ...In this paper,we study the quasi-stationarity and quasi-ergodicity of general Markov processes.We show,among other things,that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t,x,y)(with respect to m)which is bounded in(x,y)for every t>0,then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution.We also present several classes of Markov processes satisfying the above conditions.展开更多
In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is hon...In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is honest and thus unique. We then further show that this honest Feller minimum process is not only positive recurrent but also strongly ergodic. The generating function of the important stationary distribution is explicitly expressed. For the interest of comparison and completeness, the results of the Markov branching processes with killing and instantaneous resurrection are also briefly stated. A new result regarding strong ergodicity of this difficult case is presented. The birth and death process with killing and resurrection together with another example is also analyzed.展开更多
We consider a class of stochastic Boussinesq equations driven by L6vy processes and establish the uniqueness of its invariant measure. The proof is based on the progressive stopping time technique.
We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ（·）. We focus on the functional ergodic limits for the occupation time processes of the...We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ（·）. We focus on the functional ergodic limits for the occupation time processes of the models in IR. It is proved that the limiting process has the form of λζ（·）, where A is the Lebesgue measure on R and ζ（·） is a real-valued process which is non-degenerate if and only if cr is integrable. When ζ（·） is non-degenerate, it is strictly positive for t 〉 0. Moreover, ζ converges to 0 in finite-dimensional distributions if the integral of a tends to infinity.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11301030)Beijing Higher Education Young Elite Teacher Project
文摘In this work,for a one-dimensional regime-switching diffusion process,we show that when it is positive recurrent,then there exists a stationary distribution,and when it is null recurrent,then there exists an invariant measure. We also provide the explicit representation of the stationary distribution and invariant measure based on the hitting times of the process.
基金supported by the Simons Foundation (Grant No. 209206)a General Research Fund of the University of Kansas
文摘For a given probability density function p（x） on R^d, we construct a （non-stationary） diffusion process xt, starting at any point x in R^d, such that 1/T ∫_o^T δ（xt-x）dt converges to p（x） almost surely. The rate of this convergence is also investigated. To find this rate, we mainly use the Clark-Ocone formula from Malliavin calculus and the Girsanov transformation technique.
基金the National Natural Science Foundation of China(No.10671212)the Research Fund for the Doctoral Program of Higher Education(No.20050533036).
文摘For an ergodic continuous-time Markov process with a particular state in its space,the authors provide the necessary and sufficient conditions for exponential and strong ergodicity in terms of the moments of the first hitting time on the state.An application to the queue length process of M/G/1 queue with multiple vacations is given.
基金supported by National Natural Science Foundation of China(Grant No. 10971253)
文摘In this paper, we prove some limit theorems for killed Brownian motion during its life time. The emphases are on quasi-stationarity and quasi-ergodicity and related problems. On one hand, using an eigenfunction expansion for the transition density, we prove the existence and uniqueness of both quasi-stationary distribution (qsd) and mean ratio quasi-stationary distribution (mrqsd). The later is shown to be closely related to laws of large numbers (LLN) and to quasi-ergodicity. We further show that the mrqsd is the unique stationary distribution of a certain limiting ergodic diffusion process of the BM conditioned on not having been killed. We also show that a phase transition occurs from mrqsd to qsd. On the other hand, we study the large deviation behavior related to the above problems. A key observation is that the mrqsd is the unique minimum of certain large deviation rate function. We further prove that the limiting diffusion process also satisfies a large deviation principle with the rate function attaining its unique minimum at the mrqsd. These give interpretations of the mrqsd from different points of view, and establish some intrinsic connections among the above topics. Some general results concerning Yaglom limit, moment convergence and LLN are also obtained.
基金supported by National Natural Science Foundation of China(GrantNo.11171010)Beijing Natural Science Foundation(Grant No.1112001)
文摘In this paper,we study the quasi-stationarity and quasi-ergodicity of general Markov processes.We show,among other things,that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t,x,y)(with respect to m)which is bounded in(x,y)for every t>0,then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution.We also present several classes of Markov processes satisfying the above conditions.
基金Xiangtan University New Staff Research Start-up Grant (Grant No. 08QDZ27)
文摘In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is honest and thus unique. We then further show that this honest Feller minimum process is not only positive recurrent but also strongly ergodic. The generating function of the important stationary distribution is explicitly expressed. For the interest of comparison and completeness, the results of the Markov branching processes with killing and instantaneous resurrection are also briefly stated. A new result regarding strong ergodicity of this difficult case is presented. The birth and death process with killing and resurrection together with another example is also analyzed.
基金supported by National Natural Science Foundation of China (Grant Nos.10971225 and 11101427)Natural Science Foundation of Hunan Province (Grant No. 11JJ3004)the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State the Ministry of Education of China
文摘We consider a class of stochastic Boussinesq equations driven by L6vy processes and establish the uniqueness of its invariant measure. The proof is based on the progressive stopping time technique.
基金supported by Innovation Program of Shanghai Municipal Education Commission(Grant No.13zz037)the Fundamental Research Funds for the Central Universities
文摘We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ（·）. We focus on the functional ergodic limits for the occupation time processes of the models in IR. It is proved that the limiting process has the form of λζ（·）, where A is the Lebesgue measure on R and ζ（·） is a real-valued process which is non-degenerate if and only if cr is integrable. When ζ（·） is non-degenerate, it is strictly positive for t 〉 0. Moreover, ζ converges to 0 in finite-dimensional distributions if the integral of a tends to infinity.
