# zbMATH — the first resource for mathematics

Law of large numbers and central limit theorem for randomly forced PDE’s. (English) Zbl 1099.35188
The author considers the class of dissipative partial differential equations perturbed by an external random force. In particular it is studied an evolution equation in the space $$H$$ of divergence-free vector fields $$u\in L^2(D,\mathbb R^2)$$ whose normal component vanishes at $$\partial D$$: $$\dot u+Lu+B(u,u)=\eta(t)$$. Here $$L$$ is the Stokes operator and $$B$$ is a bilinear form. It is assumed that the right-hand side $$\eta$$ is a random process of the form $$\eta(t)=\sum_{j=1}^{\infty}b_{j}\dot\beta_{j}(t)e_{j}$$, where $$b_{j}\geq0$$ are some constants such that $$\sum_{j}b_{j}^2<\infty$$; $$e_{j}$$ is a complete set of normalized eigenfunctions of $$L$$, and $$\{\beta_{j}\}$$ is a sequence of independent standard Brownian motions.
The following statements are a simplified version of the main results of the paper. Suppose that the non-degeneracy condition $$b_{j}\neq0$$ for $$j=1,\ldots,N$$ is satisfied for a sufficiently large $$N$$. Then for any uniformly Lipschitz bounded functional $$f: H\to \mathbb R$$ and any solution $$u(t)$$ of the considered equation with deterministic initial condition the following assertions hold. For any $$\varepsilon>0$$ there is an a.s. finite random constant $$T\geq1$$ such that $\left| {1\over t}\int_{0}^{t}f(u(s))\,ds-(f,u)\right| \leq \text{const}\;t^{-1/2+\varepsilon}\;\text{ for}\;t\geq T.$ If $$(f,\mu)=0$$, then there is a constant $$\sigma\geq0$$ depending only on $$f$$ such that, for any $$\varepsilon>0$$, we have $\sup_{z\in R}\left(\theta_{\sigma}(z)\left| P\left\{{1\over\sqrt{t}}\int_{0}^{t}f(u(s))\,ds\leq z\right\}-\Phi_{\sigma}(z)\right| \right)\leq \;\text{ const}\;t^{-1/4+\varepsilon} \;\text{ for}\;t\geq1,$ where $$\theta_{\sigma}\equiv1$$ for $$\sigma>0$$, $$\theta_0(z)=1\wedge| z|$$, and $$\Phi_{\sigma}(z)$$ is the centered Gaussian distribution function with variance $$\sigma$$.

##### MSC:
 35R60 PDEs with randomness, stochastic partial differential equations 60F05 Central limit and other weak theorems 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35Q30 Navier-Stokes equations
Full Text:
##### References:
 [1] Bolthausen, E.: The Berry-Esseén theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 (3), 283–289 (1982) · Zbl 0492.60026 [2] Bolthausen, E.: Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 (3), 672–688 (1982) · Zbl 0494.60020 [3] Bricmont, J., Kupiainen, A., Lefevere, R.: Probabilistic estimates for the two-dimensional Navier–Stokes equations. J. Statist. Phys. 100 (3–4), 743–756 (2000) · Zbl 0972.60044 [4] Bricmont, J., Kupiainen, A., Lefevere, R.: Exponential mixing for the 2D stochastic Navier–Stokes dynamics. Commun. Math. Phys. 230 (1), 87–132 (2002) · Zbl 1033.76011 [5] Bricmont, J.: Ergodicity and mixing for stochastic partial differential equations. Proceedings of the International Congress of Mathematicians. Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 567–585 · Zbl 1031.35154 [6] Da Prato, G., Zabczyk, J.: Ergodicity for Infinite–Dimensional Systems. Cambridge University Press, Cambridge, 1996 · Zbl 0849.60052 [7] E, W., Mattingly, J.C., Sinai, Ya.G.: Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun. Math. Phys. 224 (1), 83–106 (2001) · Zbl 0994.60065 [8] Eckmann, J.-P., Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Commun. Math. Phys. 219 (3), 523–565 (2001) · Zbl 0983.60058 [9] Ferrario, B.: Ergodic results for stochastic Navier-Stokes equation. Stochastics Stochastics Rep. 60 (3–4), 271–288 (1997) · Zbl 0882.60059 [10] Flandoli, F., Maslowski, B.: Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Commun. Math. Phys. 172 (1), 119–141 (1995) · Zbl 0845.35080 [11] Gallavotti, G.: Foundations of Fluid Dynamics. Springer-Verlag, Berlin, 2002 · Zbl 0986.76001 [12] Hairer, M.: Exponential mixing properties of stochastic PDE’s through asymptotic coupling. Probab. Theory Relat. Fields 124 (3), 345–380 (2002) · Zbl 1032.60056 [13] Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. Academic Press, New York–London, 1980 · Zbl 0462.60045 [14] Hasminskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Teor. Verojatnost. i Primenen. 5, 196–214 (1960) · Zbl 0093.14902 [15] Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin, 1987 · Zbl 0635.60021 [16] Kuksin, S.: On exponential convergence to a stationary measure for nonlinear PDE’s, perturbed by random kick-forces, and the turbulence-limit. The M.I. Vishik Moscow PDE seminar, AMS Translations, 2002 [17] Kuksin, S.: Ergodic theorems for 2D statistical hydrodynamics. Rev. Math. Physics 14 (6), 585–600 (2002) · Zbl 1030.37054 [18] Kuksin, S., Shirikyan, A.: Stochastic dissipative PDE’s and Gibbs measures. Commun. Math. Phys. 213 (2), 291–330 (2000) · Zbl 0974.60046 [19] Kuksin, S., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDE’s. I. Commun. Math. Phys. 221 (2), 351–366 (2001) · Zbl 0991.60056 [20] Kuksin, S., Piatnitski, A., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDE’s. II. Commun. Math. Phys. 230 (1), 81–85 (2002) · Zbl 1010.60066 [21] Kuksin, S., Shirikyan, A.: Coupling approach to white-forced nonlinear PDE’s. J. Math. Pures Appl. 81 (6), 567–602 (2002) · Zbl 1021.37044 [22] Kuksin, S., Shirikyan, A.: Some limiting properties of randomly forced 2D Navier–Stokes equations. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), 875–891 (2003) · Zbl 1060.35106 [23] Landers, D., Rogge, L.: On the rate of convergence in the central limit theorem for Markov-chains. Z. Wahrsch. Verw. Gebiete 35 (1), 57–63 (1976) · Zbl 0315.60014 [24] Liptser, R.S., Shiryayev, A.N.: Theory of Martingales. Kluwer, Dordrecht, 1989 · Zbl 0728.60048 [25] Maruyama, G., Tanaka, H.: Some properties of one-dimensional diffusion processes. Mem. Fac. Sci. Kyusyu Univ. Ser. A. Math. 11, 117–141 (1957) · Zbl 0089.34604 [26] Maruyama, G., Tanaka, H.: Ergodic property of N-dimensional recurrent Markov processes. Mem. Fac. Sci. Kyushu Univ. Ser. A 13, 157–172 (1959) · Zbl 0115.36003 [27] Masmoudi, N., Young, L.-S.: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Commun. Math. Phys. 227 (3), 461–481 (2002) · Zbl 1009.37049 [28] Mattingly, J.C.: Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity. Commun. Math. Phys. 206 (2), 273–288 (1999) · Zbl 0953.37023 [29] Mattingly, J.C.: Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230 (3), 421–462 (2002) · Zbl 1054.76020 [30] Mattingly, J.C.: On recent progress for the stochastic Navier Stokes equations. Journées ”Équations aux Dérivées Partielles”, Exp. No. XI, 52 pp., Univ. Nantes, Nantes, 2003 [31] Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer-Verlag London, London, 1993 · Zbl 0925.60001 [32] Rio, E.: Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Springer, Berlin–Heidelberg, 2000 [33] Sawyer, S.: Rates of convergence for some functionals in probability. Ann. Math. Statist. 43, 273–284 (1972) · Zbl 0239.60024 [34] Shirikyan, A.: Analyticity of solutions for randomly forced two-dimensional Navier-Stokes equations. Russian Math. Surveys 57 (4), 785–799 (2002) · Zbl 1051.35050 [35] Shirikyan, A.: A version of the law of large numbers and applications. In: Probabilistic Methods in Fluids, Proceedings of the Swansea Workshop held on 14 – 19 April 2002, I.M. Davies et al (eds.), World Scientific, New Jersey, 2003, pp. 263–272 · Zbl 1066.76020 [36] Shirikyan, A.: Exponential mixing for 2D Navier–Stokes equations perturbed by an unbounded noise. J. Math. Fluid Mech. 6 (2), 169–193 (2004) · Zbl 1095.35032 [37] Shirikyan, A.: Some mathematical problems of statistical hydrodynamics. In: Proceedings of the International Congress of Mathematical Physics, Lisbon, 2003. To appear · Zbl 1066.76020 [38] Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam–New York–Oxford, 1977 · Zbl 0383.35057 [39] Veretennikov, A.Yu.: Estimates of the mixing rate for stochastic equations. Theory Probab. Appl. 32 (2), 273–281 (1987) · Zbl 0663.60046 [40] Vishik, M.I., Fursikov, A.V.: Mathematical Problems in Statistical Hydromechanics. Kluwer, Dordrecht, 1988 · Zbl 0688.35077 [41] Watanabe, H., Hisao, M.: Ergodic property of recurrent diffusion processes. J. Math. Soc. Japan 10, 272–286 (1958) · Zbl 0089.34701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.