The Annals of Probability

Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations

Martin Hairer and Jonathan C. Mattingly

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Abstract

We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an $\L^{p}$-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1-Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared toward total variation convergence, often fail to hold. In the first part of this paper, we consider semigroups that have uniform behavior which one can view as the analog of Doeblin’s condition. We then proceed to study situations where the behavior is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier–Stokes equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show that the stochastic Navier–Stokes equations’ invariant measures depend continuously on the viscosity and the structure of the forcing.

Article information

Source
Ann. Probab. Volume 36, Number 6 (2008), 2050-2091.

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.aop/1229696596

Digital Object Identifier
doi:10.1214/08-AOP392

Mathematical Reviews number (MathSciNet)
MR2478676

Zentralblatt MATH identifier
1173.37005

Subjects
Primary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 37A25: Ergodicity, mixing, rates of mixing 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
Stochastic PDEs Wasserstein distance ergodicity mixing spectral gap

Citation

Hairer, Martin; Mattingly, Jonathan C. Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36 (2008), no. 6, 2050--2091. doi:10.1214/08-AOP392. http://projecteuclid.org/euclid.aop/1229696596.


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References

  • [1] Agrachëv, A. A. and Sarychev, A. V. (2004). Controllability for the Navier–Stokes equation with small control. Dokl. Akad. Nauk 394 727–730.
  • [2] Agrachev, A. A. and Sarychev, A. V. (2005). Navier–Stokes equations: Controllability by means of low modes forcing. J. Math. Fluid Mech. 7 108–152.
  • [3] Bakhtin, Y. and Mattingly, J. C. (2007). Malliavin calculus for infinite-dimensional systems with additive noise. J. Funct. Anal. 249 307–353.
  • [4] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités XIX, 1983/84. Lecture Notes in Math. 1123 177–206. Springer, Berlin.
  • [5] Bricmont, J., Kupiainen, A. and Lefevere, R. (2001). Ergodicity of the 2D Navier–Stokes equations with random forcing. Comm. Math. Phys. 224 65–81. Dedicated to Joel L. Lebowitz.
  • [6] Bricmont, J., Kupiainen, A. and Lefevere, R. (2002). Exponential mixing of the 2D stochastic Navier–Stokes dynamics. Comm. Math. Phys. 230 87–132.
  • [7] Constantin, P. and Foias, C. (1988). Navier–Stokes Equations. Chicago Lectures in Mathematics. Univ. of Chicago Press, Chicago, IL.
  • [8] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications 44. Cambridge Univ. Press, Cambridge.
  • [9] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Press, Cambridge.
  • [10] Davies, E. B. (1980). One-Parameter Semigroups. London Mathematical Society Monographs 15. Academic Press [Harcourt Brace Jovanovich Publishers], London.
  • [11] Doeblin, W. (1937). Sur les propriétés asymptotiques de mouvement régis par certains types de chaînes simples. Bull. Math. Soc. Roum. Sci. 39 57–115.
  • [12] Doob, J. L. (1948). Asymptotic properties of Markoff transition prababilities. Trans. Amer. Math. Soc. 63 393–421.
  • [13] E, W. and Mattingly, J. C. (2001). Ergodicity for the Navier–Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math. 54 1386–1402.
  • [14] E, W., Mattingly, J. C. and Sinai, Y. (2001). Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Comm. Math. Phys. 224 83–106. Dedicated to Joel L. Lebowitz.
  • [15] Eckmann, J.-P. and Hairer, M. (2001). Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Comm. Math. Phys. 219 523–565.
  • [16] Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge Univ. Press, Cambridge.
  • [17] Flandoli, F. and Maslowski, B. (1995). Ergodicity of the 2-D Navier–Stokes equation under random perturbations. Comm. Math. Phys. 172 119–141.
  • [18] Foiaş, C. and Prodi, G. (1967). Sur le comportement global des solutions nonstationnaires des équations de Navier–Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova 39 1–34.
  • [19] Goldys, B. and Maslowski, B. (2005). Exponential ergodicity for stochastic Burgers and 2D Navier–Stokes equations. J. Funct. Anal. 226 230–255.
  • [20] Hairer, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 345–380.
  • [21] Hairer, M. and Mattingly, J. C. (2006). Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 993–1032.
  • [22] Harris, T. E. (1956). The existence of stationary measures for certain Markov processes. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 1954–1955 II 113–124. Univ. California Press, Berkeley.
  • [23] Hennion, H. (1993). Sur un théorème spectral et son application aux noyaux Lipchitziens. Proc. Amer. Math. Soc. 118 627–634.
  • [24] Hörmander, L. (1967). Hypoelliptic second order differential equations. Acta Math. 119 147–171.
  • [25] Huang, J., Kontoyiannis, I. and Meyn, S. P. (2002). The ODE method and spectral theory of Markov operators. In Stochastic Theory and Control (Lawrence, KS, 2001). Lecture Notes in Control and Inform. Sci. 280 205–221. Springer, Berlin.
  • [26] Ionescu Tulcea, C. T. and Marinescu, G. (1950). Théorie ergodique pour des classes d’opérations noncomplètement continues. Ann. of Math. (2) 52 140–147.
  • [27] Kuksin, S. and Shirikyan, A. (2001). Ergodicity for the randomly forced 2D Navier–Stokes equations. Math. Phys. Anal. Geom. 4 147–195.
  • [28] Kuksin, S. and Shirikyan, A. (2001). A coupling approach to randomly forced nonlinear PDE’s. I. Comm. Math. Phys. 221 351–366.
  • [29] Kuksin, S. and Shirikyan, A. (2002). Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. (9) 81 567–602.
  • [30] Lasota, A. and Yorke, J. A. (1973). On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 481–488 (1974).
  • [31] Liverani, C. (2003). Invariant measures and their properties. A functional analytic point of view. In Dynamical Systems. Part II. Pubbl. Cent. Ric. Mat. Ennio Giorgi 185–237. Scuola Norm. Sup., Pisa.
  • [32] Majda, A. and Wang, X. (2006). The emergence of large-scale coherent structure under small-scale random bombardments. Comm. Pure Appl. Math. 59 467–500.
  • [33] Malliavin, P. (1978). Stochastic calculus of variation and hypoelliptic operators. In Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) 195–263. Wiley, New York.
  • [34] Masmoudi, N. and Young, L.-S. (2002). Ergodic theory of infinite-dimensional systems with applications to dissipative parabolic PDEs. Comm. Math. Phys. 227 461–481.
  • [35] Mattingly, J. C. (1998). The stochastically forced Navier–Stokes equations: Energy estimates and phase space contraction. Ph.D. thesis, Princeton Univ.
  • [36] Mattingly, J. C. (1999). Ergodicity of 2D Navier–Stokes equations with random forcing and large viscosity. Comm. Math. Phys. 206 273–288.
  • [37] Mattingly, J. C. (2002). Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 421–462.
  • [38] Mattingly, J. C. (2003). On recent progress for the stochastic Navier–Stokes equations. In Journées “Équations aux Dérivées Partielles” Exp. No. XI, 52. Univ. Nantes, Nantes.
  • [39] Mattingly, J. C. and Pardoux, É. (2006). Malliavin calculus for the stochastic 2D Navier–Stokes equation. Comm. Pure Appl. Math. 59 1742–1790.
  • [40] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer London Ltd., London.
  • [41] Norris, J. (1986). Simplified Malliavin calculus. In Séminaire de Probabilités XX, 1984/85. Lecture Notes in Math. 1204 101–130. Springer, Berlin.
  • [42] Nussbaum, R. D. (1970). The radius of the essential spectrum. Duke Math. J. 37 473–478.
  • [43] Odasso, C. (2006). Ergodicity for the stochastic complex Ginzburg–Landau equations. Ann. Inst. H. Poincaré Probab. Statist. 42 417–454.
  • [44] Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, Chichester.
  • [45] Robinson, J. C. (2002). Stability of random attractors under perturbation and approximation. J. Differential Equations 186 652–669.
  • [46] Röckner, M. and Sobol, Z. (2004). A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations. C. R. Math. Acad. Sci. Paris 338 945–949.
  • [47] Romito, M. (2004). Ergodicity of the finite-dimensional approximation of the 3D Navier–Stokes equations forced by a degenerate noise. J. Statist. Phys. 114 155–177.
  • [48] Stroock, D. W. (1981). The Malliavin calculus and its applications. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math. 851 394–432. Springer, Berlin.
  • [49] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
  • [50] Villani, C. (2008). Optimal Transport, Old and New. Saint Flour Lectures. Springer.