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
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.
