%0 Journal Article %1 shirikyan2018controllability %A Shirikyan, Armen %D 2018 %K SPDE controllability ergodicity %R 10.1070/RM9755 %T Controllability implies mixing I. Convergence in the total variation metric %U http://arxiv.org/abs/1803.01892 %X This paper is the first part of a project devoted to studying the interconnection between controllability properties of a dynamical system and the large-time asymptotics of trajectories for the associated stochastic system. It is proved that the approximate controllability to a given point and the solid controllability from the same point imply the uniqueness of a stationary measure and exponential mixing in the total variation metric. This result is then applied to random differential equations on a compact Riemannian manifold. In the second part, we shall replace the solid controllability by a stabilisability condition and prove that it is still sufficient for the uniqueness of a stationary distribution, whereas the convergence to it holds in the weaker dual-Lipschitz metric.

@article{shirikyan2018controllability, abstract = {This paper is the first part of a project devoted to studying the interconnection between controllability properties of a dynamical system and the large-time asymptotics of trajectories for the associated stochastic system. It is proved that the approximate controllability to a given point and the solid controllability from the same point imply the uniqueness of a stationary measure and exponential mixing in the total variation metric. This result is then applied to random differential equations on a compact Riemannian manifold. In the second part, we shall replace the solid controllability by a stabilisability condition and prove that it is still sufficient for the uniqueness of a stationary distribution, whereas the convergence to it holds in the weaker dual-Lipschitz metric.}, added-at = {2018-03-15T15:51:04.000+0100}, author = {Shirikyan, Armen}, biburl = {https://www.bibsonomy.org/bibtex/2bf4bfac6cdf480bf1bd5ef156847c7b4/claired}, description = {Controllability implies mixing I. Convergence in the total variation metric}, doi = {10.1070/RM9755}, interhash = {37259029bc2f953e68b8c6de362b055b}, intrahash = {bf4bfac6cdf480bf1bd5ef156847c7b4}, keywords = {SPDE controllability ergodicity}, note = {cite arxiv:1803.01892}, timestamp = {2018-03-15T15:51:04.000+0100}, title = {Controllability implies mixing I. Convergence in the total variation metric}, url = {http://arxiv.org/abs/1803.01892}, year = 2018 }