We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces.More preci...We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces.More precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous MarkovFeller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior,then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the stochastically continuous continuous-time Markov-Feller semigroups.展开更多
This work focuses on a class of jump-diffusions with state-dependent switching. First, compared with the existing results in the literature, in our model, the characteristic measure is allowed to be a-finite. The exis...This work focuses on a class of jump-diffusions with state-dependent switching. First, compared with the existing results in the literature, in our model, the characteristic measure is allowed to be a-finite. The existence and uniqueness of the underlying process are obtained by representing the switching component as a stochastic integral with respect to a Poisson random measure and by using a successive approximation method. Then, the Feller property is proved by means of introducing auxiliary processes and by making use of Radon-Nikodym derivatives. Furthermore, the irreducibility and all compact sets being petite are demonstrated. Based on these results, the uniform ergodicity is established under a general Lyapunov condition. Finally, easily verifiable conditions for uniform ergodicity are established when the jump-diffusions are linearizable with respect to the variable x （the state variable corresponding to the jump-diffusion component） in a neighborhood of the infinity, and some examples are presented to illustrate the results.展开更多
A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properti...A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properties for the extended birth_death processes with catastrophes are obtained.展开更多
We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitabl...We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions Xt: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (Xt,Λ(t));and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.展开更多
In this paper, by using a semimartingale approximation of a fractional stochastic integration, the global Harnack inequalities for stochastic retarded differential equations driven by fractional Brownian motion with H...In this paper, by using a semimartingale approximation of a fractional stochastic integration, the global Harnack inequalities for stochastic retarded differential equations driven by fractional Brownian motion with Hurst parameter 0 〈 H 〈 1 are established. As applications, strong Feller property, log-Harnack inequality and entropycost inequality are given.展开更多
We investigate periodic solutions of regime-switching jump diffusions.We first show the well-posedness of solutions to stochastic differential equations corresponding to the hybrid system.Then,we derive the strong Fel...We investigate periodic solutions of regime-switching jump diffusions.We first show the well-posedness of solutions to stochastic differential equations corresponding to the hybrid system.Then,we derive the strong Feller property and irreducibility of the associated time-inhomogeneous semigroups.Finally,we establish the existence and uniqueness of periodic solutions.Concrete examples are presented to illustrate the results.展开更多
This work focuses on stochastic Lienard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property i...This work focuses on stochastic Lienard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly.展开更多
Consider the two-dimensional, incompressible Navier-Stokes equations on torus T^2= [-π, π]^2 driven by a degenerate multiplicative noise in the vorticity formulation（abbreviated as SNS）： dwt = ν？w_tdt ＋B（Kw_t...Consider the two-dimensional, incompressible Navier-Stokes equations on torus T^2= [-π, π]^2 driven by a degenerate multiplicative noise in the vorticity formulation（abbreviated as SNS）： dwt = ν？w_tdt ＋B（Kw_t, w_t）dt ＋ Q（w_t）dW t. We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup{P_t}_t≥0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility.Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140： 41–82（2005）with a different method, we get an exponential ergodicity under a stronger norm.展开更多
基金supported by National Natural Science Foundation of China (No.11731009, No.12231002)Center for Statistical Science,Peking University。
文摘We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces.More precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous MarkovFeller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior,then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the stochastically continuous continuous-time Markov-Feller semigroups.
基金supported in part by National Natural Science Foundation of China(Grant No. 11171024)supported in part by National Natural Science Foundation of China (Grant No.70871055)supported in part by National Science Foundationof US (Grant No. DMS-0907753)
文摘This work focuses on a class of jump-diffusions with state-dependent switching. First, compared with the existing results in the literature, in our model, the characteristic measure is allowed to be a-finite. The existence and uniqueness of the underlying process are obtained by representing the switching component as a stochastic integral with respect to a Poisson random measure and by using a successive approximation method. Then, the Feller property is proved by means of introducing auxiliary processes and by making use of Radon-Nikodym derivatives. Furthermore, the irreducibility and all compact sets being petite are demonstrated. Based on these results, the uniform ergodicity is established under a general Lyapunov condition. Finally, easily verifiable conditions for uniform ergodicity are established when the jump-diffusions are linearizable with respect to the variable x （the state variable corresponding to the jump-diffusion component） in a neighborhood of the infinity, and some examples are presented to illustrate the results.
文摘A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properties for the extended birth_death processes with catastrophes are obtained.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.12071031).
文摘We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions Xt: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (Xt,Λ(t));and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.
基金This research is partially supported by the NNSF of China （No. 61273179） and Natural Science Foundation of Hubei Province （No. 2016CFB479）.
文摘In this paper, by using a semimartingale approximation of a fractional stochastic integration, the global Harnack inequalities for stochastic retarded differential equations driven by fractional Brownian motion with Hurst parameter 0 〈 H 〈 1 are established. As applications, strong Feller property, log-Harnack inequality and entropycost inequality are given.
基金The authors thank the referees for the careful reading of their paper and all of the insightful suggestions and comments that greatly improved the presentation of the paper.This work was supported by the research fund from Shanxi Province Department of Finance and Education for Ph.D.Graduates to Work in Shanxi(No.2021-18,125/Z24179)and the Natural Sciences and Engineering Research Council of Canada(No.4394-2018).
文摘We investigate periodic solutions of regime-switching jump diffusions.We first show the well-posedness of solutions to stochastic differential equations corresponding to the hybrid system.Then,we derive the strong Feller property and irreducibility of the associated time-inhomogeneous semigroups.Finally,we establish the existence and uniqueness of periodic solutions.Concrete examples are presented to illustrate the results.
基金Supported by the National Natural Science Foundation of China(No.11171024)the National Science Foundation,United States(No.DMS-0907753)
文摘This work focuses on stochastic Lienard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly.
基金supported by the National Natural Science Foundation of China(No.11371041,11431014)the Key Laboratory of Random Complex Structures and Data Science,Academy of Mathematics and Systems Science,Chinese Academy of Sciences(No.2008DP173182)+3 种基金supported by NSFC(No.11501195)a Scientific Research Fund of Hunan Provincial Education Department(No.17C0953)the Youth Scientific Research Fund of Hunan Normal University(No.Math140650)the Construct Program of the Key Discipline in Hunan Province
文摘Consider the two-dimensional, incompressible Navier-Stokes equations on torus T^2= [-π, π]^2 driven by a degenerate multiplicative noise in the vorticity formulation（abbreviated as SNS）： dwt = ν？w_tdt ＋B（Kw_t, w_t）dt ＋ Q（w_t）dW t. We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup{P_t}_t≥0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility.Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140： 41–82（2005）with a different method, we get an exponential ergodicity under a stronger norm.
基金supported by the Youth Scientific Research Fund of Hunan Normal.University(No.Math140650)the Scientific Research Foundation for Ph.D Hunan Normal University(No.Math140675)
