In this paper, we focus on the space-inhomogeneous three-state on the one-dimension lattice, a one-phase model and a two-phase model include. By using the transfer matrices method by Endo et al., we calculate the stat...In this paper, we focus on the space-inhomogeneous three-state on the one-dimension lattice, a one-phase model and a two-phase model include. By using the transfer matrices method by Endo et al., we calculate the stationary measure for initial state concrete eigenvalue. Finally we found the transfer matrices method is more effective for the three-state quantum walks than the method obtained by Kawai et al.展开更多
The limiting behavior of stochastic evolution processes with small noise intensityεis investigated in distribution-based approaches.Letμεbe a stationary measure for stochastic process Xεwith smallεand X0 be a sem...The limiting behavior of stochastic evolution processes with small noise intensityεis investigated in distribution-based approaches.Letμεbe a stationary measure for stochastic process Xεwith smallεand X0 be a semiflow on a Polish space.Assume that{με:0<ε≤ε0}is tight.Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0.Applications are made to various stochastic evolution systems,including stochastic ordinary differential equations,stochastic partial differential equations,and stochastic functional differential equations driven by Brownian motion or Levy processes.展开更多
This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a n...This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a null set of any limit measure.Using this criterion,we first provide a series of nontrivial concrete examples to show that their repelling limit cycles or quasi-periodic orbits are null sets for all limit measures,which deduces that all their limit measures are concentrated on stable equilibria and stable limit cycles or quasi-periodic orbits,and saddles.Interesting open questions on exact supports of limit measures are proposed.展开更多
The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic di erential equations(SDEs)with less regular coecients and degenerate noises.These equations are often ...The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic di erential equations(SDEs)with less regular coecients and degenerate noises.These equations are often derived as mesoscopic limits of complex or huge microscopic systems.By studying the associated Fokker-Planck equation(FPE),we prove the convergence of the time average of globally de ned weak solutions of such an SDE to the set of stationary measures of the FPE under Lyapunov conditions.In the case where the set of stationary measures consists of a single element,the unique stationary measure is shown to be physical.Similar convergence results for the solutions of the FPE are established as well.Some of our convergence results,while being special cases of those contained in Ji et al.(2019)for SDEs with periodic coecients,have weaken the required Lyapunov conditions and are of much simpli ed proofs.Applications to stochastic damping Hamiltonian systems and stochastic slow-fast systems are given.展开更多
文摘In this paper, we focus on the space-inhomogeneous three-state on the one-dimension lattice, a one-phase model and a two-phase model include. By using the transfer matrices method by Endo et al., we calculate the stationary measure for initial state concrete eigenvalue. Finally we found the transfer matrices method is more effective for the three-state quantum walks than the method obtained by Kawai et al.
基金supported by National Natural Science Foundation of China(Grant Nos.11771295,11271356,11371041,11431014 and 11401557)Key Laboratory of Random Complex Structures and Data Science,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,and the Fundamental Research Funds for the Central Universities(Grant No.WK0010000048)。
文摘The limiting behavior of stochastic evolution processes with small noise intensityεis investigated in distribution-based approaches.Letμεbe a stationary measure for stochastic process Xεwith smallεand X0 be a semiflow on a Polish space.Assume that{με:0<ε≤ε0}is tight.Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0.Applications are made to various stochastic evolution systems,including stochastic ordinary differential equations,stochastic partial differential equations,and stochastic functional differential equations driven by Brownian motion or Levy processes.
基金supported by National Natural Science Foundation of China(Grant Nos.11771295,11431014,11931004 and 11371252)Key Laboratory of Random Complex Structures and Data Science,Academy of Mathematics and Systems Science,Chinese Academy of Sciences(Grant No.2008DP173182)。
文摘This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a null set of any limit measure.Using this criterion,we first provide a series of nontrivial concrete examples to show that their repelling limit cycles or quasi-periodic orbits are null sets for all limit measures,which deduces that all their limit measures are concentrated on stable equilibria and stable limit cycles or quasi-periodic orbits,and saddles.Interesting open questions on exact supports of limit measures are proposed.
基金The first author was supported by China Scholarship Council.The second author was supported by University of Alberta,and Natural Sciences and Engineering Research Council of Canada(Grant Nos.RGPIN-2018-04371 and DGECR-2018-00353)The third author was supported by Pacific Institute for the Mathematical Sciences-Canadian Statistical Sciences Institute Postdoctoral Fellowship,Pacific Institute for the Mathematical Sciences-Collaborative Research Group Grant,National Natural Science Foundation of China(Grant Nos.11771026 and 11471344)+2 种基金the Pacific Institute for the Mathematical Sciences-University of Washington site through National Science Foundation of USA(Grant No.DMS-1712701)The fourth author was supported by Natural Sciences and Engineering Research Council of Canada Discovery(Grant No.1257749)Pacific Institute for the Mathematical Sciences-Collaborative Research Group Grant,University of Alberta,and Jilin University.
文摘The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic di erential equations(SDEs)with less regular coecients and degenerate noises.These equations are often derived as mesoscopic limits of complex or huge microscopic systems.By studying the associated Fokker-Planck equation(FPE),we prove the convergence of the time average of globally de ned weak solutions of such an SDE to the set of stationary measures of the FPE under Lyapunov conditions.In the case where the set of stationary measures consists of a single element,the unique stationary measure is shown to be physical.Similar convergence results for the solutions of the FPE are established as well.Some of our convergence results,while being special cases of those contained in Ji et al.(2019)for SDEs with periodic coecients,have weaken the required Lyapunov conditions and are of much simpli ed proofs.Applications to stochastic damping Hamiltonian systems and stochastic slow-fast systems are given.
