Geodesic
Controllability implies mixing. I. Convergence in the total variation metric
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 5, pp. 939-953 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper is the first part of a project to study the interconnection between the 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 of the project, the solid controllability will be replaced by a stabilisability condition, and it will be proved that this is still sufficient for the uniqueness of a stationary distribution, whereas the convergence to it occurs in the weaker dual-Lipschitz metric. Bibliography: 21 titles.
Keywords: controllability, ergodicity, exponential mixing. 
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A. R. Shirikyan. Controllability implies mixing. I. Convergence in the total variation metric. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 5, pp. 939-953. http://geodesic.mathdoc.fr/item/RM_2017_72_5_a3/

[1] A. Agrachev, S. Kuksin, A. Sarychev, A. Shirikyan, “On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier–Stokes equations”, Ann. Inst. H. Poincaré Probab. Statist., 43:4 (2007), 399–415 | DOI | MR | Zbl

[2] A. A. Agrachev, Yu. L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia Math. Sci., 87, Control Theory and Optimization, II, Springer-Verlag, Berlin, 2004, xiv+412 pp. | DOI | MR | Zbl

[3] A. A. Agrachev, A. V. Sarychev, “Navier–Stokes equations: controllability by means of low modes forcing”, J. Math. Fluid Mech., 7:1 (2005), 108–152 | DOI | MR | Zbl

[4] A. Agrachev, A. Sarychev, “Solid controllability in fluid dynamics”, Instability in models connected with fluid flows. I, Int. Math. Ser. (N. Y.), 6, Springer, New York, 2008, 1–35 | DOI | MR | Zbl

[5] L. Arnold, W. Kliemann, “On unique ergodicity for degenerate diffusions”, Stochastics, 21:1 (1987), 41–61 | DOI | MR | Zbl

[6] V. I. Bogachev, Differentiable measures and the Malliavin calculus, Math. Surveys Monogr., 164, Amer. Math. Soc., Providence, RI, 2010, xvi+488 pp. ; V. I. Bogachev, Differentsiruemye mery i ischislenie Mallyavena, In-t kompyut. issled., R Dynamics, M.–Izhevsk, 2008, 543 pp. | DOI | MR | Zbl

[7] G. Da Prato, Introduction to stochastic analysis and Malliavin calcrulus, Appunti. Sc. Norm. Super. Pisa (N. S.), 13, 3rd ed., Edizioni della Normale, Pisa, 2014, xviii+279 pp. | DOI | MR | Zbl

[8] R. M. Dudley, Real analysis and probability, Cambridge Stud. Adv. Math., 74, 2nd ed., Cambridge Univ. Press, Cambridge, 2002, x+555 pp. | DOI | MR | Zbl

[9] M. Hairer, “On Malliavin's proof of Hörmander's theorem”, Bull. Sci. Math., 135:6-7 (2011), 650–666 | DOI | MR | Zbl

[10] M. Hairer, J. C. Mattingly, “Yet another look at Harris' ergodic theorem for Markov chains”, Seminar on stochastic analysis, random fields and applications VI, Progr. Probab., 63, Birkhäuser/Springer Basel AG, Basel, 2011, 109–117 | DOI | MR | Zbl

[11] L. Hörmander, The analysis of linear partial differential operators, Classics Math., III, Pseudo-differential operators, Springer, Berlin, 2007, viii+525 pp. ; L. Khermander, Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 3, Psevdodifferentsialnye operatory, Mir, M., 1987, 694 pp. | DOI | MR | Zbl | MR

[12] V. Jurdjevic, Geometric control theory, Cambridge Stud. Adv. Math., 52, Cambridge Univ. Press, Cambridge, 1997, xviii+492 pp. | MR | Zbl

[13] R. Z. Khas'minskii, Stochastic stability of differential equations, Monographs Textbooks Mech. Solids Fluids: Mech. Analysis, 7, Sijthoff Noordhoff, Alphen aan den Rijn–Germantown, MD, 1980, xvi+344 pp. | MR | MR | Zbl | Zbl

[14] S. Kuksin, A. Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Math., 194, Cambridge Univ. Press, Cambridge, 2012, xvi+320 pp. | DOI | MR | Zbl

[15] S. P. Meyn, R. L. Tweedie, Markov chains and stochastic stability, Comm. Control Engrg. Ser., Springer-Verlag London, Ltd., London, 1993, xvi+548 pp. | DOI | MR | Zbl

[16] D. Nualart, The Malliavin calculus and related topics, Probab. Appl. (N. Y.), Springer-Verlag, New York, 1995, xii+266 pp. | DOI | MR | Zbl

[17] E. Nummelin, General irreducible Markov chains and non-negative operators, Cambridge Tracts in Math., 83, Cambridge Univ. Press, Cambridge, 1984, xi+156 pp. | DOI | MR | MR | Zbl

[18] A. Shirikyan, “Exact controllability in projections for three-dimensional Navier–Stokes equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24:4 (2007), 521–537 | DOI | MR | Zbl

[19] A. Shirikyan, “Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations”, J. Funct. Anal., 249:2 (2007), 284–306 | DOI | MR | Zbl

[20] S. Sternberg, Lectures on differential geometry, 2nd ed., Chelsea Publishing Co., New York, 1983, xviii+442 pp. | MR | MR | Zbl | Zbl

[21] A. Yu. Veretennikov, “On rate of mixing and the averaging principle for hypoelliptic stochastic differential equations”, Math. USSR-Izv., 33:2 (1989), 221–231 | DOI | MR | Zbl