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Synchronisation of almost all trajectories of a random dynamical system

The author wishes gratefully to acknowledge funding from an EPSRC Doctoral Training Account and an EPSRC Doctoral Prize, both at Imperial College London

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  • It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally asymptotically stable trajectories, then there is a random attracting set consisting of finitely many points, whose basin of forward-time attraction includes a random full measure open set. In this paper, we present necessary and sufficient conditions for this attracting set to be a singleton. Our result does not require the state space to be compact, but holds on general Lusin metric spaces (in both discrete and continuous time).

    Mathematics Subject Classification: Primary: 37H05; Secondary: 60F15.


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