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
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.