## Electronic Journal of Probability

### A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs

#### Abstract

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator $M(t)$ can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection $\Pi$ on a subspace of sufficiently regular functions. Then the eigenfunctions of $M(t)$ with small eigenvalues have only a very small component in the image of $\Pi$."

We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced by Hairer and Mattingly in Ann. of Math. (2) 164 (2006).

One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context.

We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 23, 658-738.

Dates
Accepted: 30 March 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820192

Digital Object Identifier
doi:10.1214/EJP.v16-875

Mathematical Reviews number (MathSciNet)
MR2786645

Zentralblatt MATH identifier
1228.60072

Rights

#### Citation

Hairer, Martin; Mattingly, Jonathan. A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs. Electron. J. Probab. 16 (2011), paper no. 23, 658--738. doi:10.1214/EJP.v16-875. https://projecteuclid.org/euclid.ejp/1464820192

#### References

• Agrachev, A.; Kuksin, S.; Sarychev, A.; Shirikyan, A. On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 4, 399–415.
• Agrachev, Andrei A.; Sachkov, Yuri L. Control theory from the geometric viewpoint. Encyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer-Verlag, Berlin, 2004. xiv+412 pp. ISBN: 3-540-21019-9.
• Agrachev, Andrey A.; Sarychev, Andrey V. Navier-Stokes equations: controllability by means of low modes forcing. J. Math. Fluid Mech. 7 (2005), no. 1, 108–152.
• Agrachev, Andrey; Sarychev, Andrey. Solid controllability in fluid dynamics. Instability in models connected with fluid flows. I, 1–35, Int. Math. Ser. (N. Y.), 6, Springer, New York, 2008.
• Baudoin, Fabrice; Hairer, Martin. A version of Hörmander's theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 (2007), no. 3-4, 373–395.
• Bricmont, J.; Kupiainen, A.; Lefevere, R. Ergodicity of the 2D Navier-Stokes equations with random forcing. Dedicated to Joel L. Lebowitz. Comm. Math. Phys. 224 (2001), no. 1, 65–81.
• Bakhtin, Yuri; Mattingly, Jonathan C. Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7 (2005), no. 5, 553–582.
• Bakhtin, Yuri; Mattingly, Jonathan C. Malliavin calculus for infinite-dimensional systems with additive noise. J. Funct. Anal. 249 (2007), no. 2, 307–353.
• Baudoin, Fabrice; Teichmann, Josef. Hypoellipticity in infinite dimensions and an application in interest rate theory. Ann. Appl. Probab. 15 (2005), no. 3, 1765–1777.
• Cerrai, Sandra. Ergodicity for stochastic reaction-diffusion systems with polynomial coefficients. Stochastics Stochastics Rep. 67 (1999), no. 1-2, 17–51.
• Constantin, Peter; Foias, Ciprian. Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. x+190 pp. ISBN: 0-226-11548-8; 0-226-11549-6.
• Dautray, Robert; Lions, Jacques-Louis. Mathematical analysis and numerical methods for science and technology. Vol. 5. Evolution problems. I. With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon. Translated from the French by Alan Craig. Springer-Verlag, Berlin, 1992. xiv+709 pp. ISBN: 3-540-50205-X; 3-540-66101-8
• Da Prato, G.; Elworthy, K. D.; Zabczyk, J. Strong Feller property for stochastic semilinear equations. Stochastic Anal. Appl. 13 (1995), no. 1, 35–45.
• Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6.
• Da Prato, G.; Zabczyk, J. Ergodicity for infinite-dimensional systems. London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. xii+339 pp. ISBN: 0-521-57900-7.
• Eckmann, J.-P.; Hairer, M. Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Comm. Math. Phys. 219 (2001), no. 3, 523–565.
• E, Weinan; Mattingly, J. C.; Sinai, Ya. Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Dedicated to Joel L. Lebowitz. Comm. Math. Phys. 224 (2001), no. 1, 83–106.
• Flandoli, Franco. Regularity theory and stochastic flows for parabolic SPDEs. Stochastics Monographs, 9. Gordon and Breach Science Publishers, Yverdon, 1995. x+79 pp. ISBN: 2-88449-045-0.
• Flandoli, Franco; Maslowski, Bohdan. Ergodicity of the $2$-D Navier-Stokes equation under random perturbations. Comm. Math. Phys. 172 (1995), no. 1, 119–141.
• Goldys, B.; Maslowski, B. Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's. Ann. Probab. 34 (2006), no. 4, 1451–1496.
• Hairer, M. Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 (2002), no. 3, 345–380.
• Hairer, M. An introduction to stochastic PDEs, coupling. Unpublished lecture notes (2009) http://www.hairer.org/Teaching.html.
• Hairer, Martin Mattingly, Jonathan C. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 (2006), no. 3, 993–1032.
• 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.
• Hairer, Martin; Majda, Andrew J. A simple framework to justify linear response theory. Nonlinearity 23 (2010), no. 4, 909–922.
• Hairer, Martin; Mattingly, Jonathan C. ; Scheutzow, M. Asymptotic coupling and a general form of harris? theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149 (2011), no. 1–2, 223-259.
• Hörmander, Lars. Hypoelliptic second order differential equations. Acta Math. 119 1967 147–171.
• Hörmander, Lars. The Analysis of Linear Partial Differential Operators 1–4 Springer, New York 1985.
• Ionescu Tulcea, C. T.; Marinescu, G. Théorie ergodique pour des classes d'opérations non complètement continues. (French) Ann. of Math. (2) 52, (1950). 140–147.
• Jurdjevic, Velimir. Geometric control theory. Cambridge Studies in Advanced Mathematics, 52. Cambridge University Press, Cambridge, 1997. xviii+492 pp. ISBN: 0-521-49502-4.
• Kato, TPerturbation Theory for Linear Operators.Springer,New York, 1980
• Kusuoka, Shigeo; Stroock, Daniel. Applications of the Malliavin calculus. I. Stochastic analysis (Katata/Kyoto, 1982), 271–306, North-Holland Math. Library, 32, North-Holland, Amsterdam, 1984.
• Kusuoka, S.; Stroock, D. Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 1–76.
• Kusuoka, S.; Stroock, D. Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 1–76.
• Kuksin, Sergei; Shirikyan, Armen. Stochastic dissipative PDEs and Gibbs measures. Comm. Math. Phys. 213 (2000), no. 2, 291–330.
• Liverani, Carlangelo. Invariant measures and their properties. A functional analytic point of view. Dynamical systems. Part II, 185–237, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003.
• Lasota, A.; Yorke, James A. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488 (1974).
• Malliavin, Paul. Stochastic calculus of variation and hypoelliptic operators. Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), pp. 195–263, Wiley, New York-Chichester-Brisbane, 1978.
• Malliavin, Paul. Stochastic analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 313. Springer-Verlag, Berlin, 1997. xii+343 pp. ISBN: 3-540-57024-1.
• Maslowski, Bohdan. Strong Feller property for semilinear stochastic evolution equations and applications. Stochastic systems and optimization (Warsaw, 1988), 210–224, Lecture Notes in Control and Inform. Sci., 136, Springer, Berlin, 1989.
• Mattingly, Jonathan C. Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 (2002), no. 3, 421–462.
• Mattingly, Jonathan C. Saint Flour Lectures Unpublished Lecture Notes 230 (2002), no. 3, 421–462. http://www.math.duke.edu/~jonm/
• Mattingly, Jonathan C.; Pardoux, Étienne. Malliavin calculus for the stochastic 2D Navier-Stokes equation. Comm. Pure Appl. Math. 59 (2006), no. 12, 1742–1790.
• Mattingly, Jonathan C.; Suidan, Toufic; Vanden-Eijnden, Eric. Simple systems with anomalous dissipation and energy cascade. Comm. Math. Phys. 276 (2007), no. 1, 189–220.
• Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6.
• Masmoudi, Nader; Young, Lai-Sang. Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Comm. Math. Phys. 227 (2002), no. 3, 461–481.
• Nagasawa, Takeyuki. Navier-Stokes flow on Riemannian manifolds. Proceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996). Nonlinear Anal. 30 (1997), no. 2, 825–832.
• Norris, James. Simplified Malliavin calculus. Séminaire de Probabilités, XX, 1984/85, 101–130, Lecture Notes in Math., 1204, Springer, Berlin, 1986.
• Nualart, David. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, New York, 1995. xii+266 pp. ISBN: 0-387-94432-X.
• Ocone, Daniel. Stochastic calculus of variations for stochastic partial differential equations. J. Funct. Anal. 79 (1988), no. 2, 288–331.
• Renardy, Michael; Rogers, Robert C. An introduction to partial differential equations. Second edition. Texts in Applied Mathematics, 13. Springer-Verlag, New York, 2004. xiv+434 pp. ISBN: 0-387-00444-0.
• Reed, Michael; Simon, Barry. Methods of modern mathematical physics. I. Functional analysis. Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. xv+400 pp. ISBN: 0-12-585050-6.
• Stroock, Daniel W. Some applications of stochastic calculus to partial differential equations. Eleventh Saint Flour probability summer school-1981 (Saint Flour, 1981), 267–382, Lecture Notes in Math., 976, Springer, Berlin, 1983.
• Taylor, Michael E. Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456.
• Triebel, Hans. Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds. Ark. Mat. 24 (1986), no. 2, 299–337.
• Triebel, Hans. Theory of function spaces. II. Monographs in Mathematics, 84. Birkhäuser Verlag, Basel, 1992. viii+370 pp. ISBN: 3-7643-2639-5.
• Temam, Roger; Wang, Shou Hong. Inertial forms of Navier-Stokes equations on the sphere. J. Funct. Anal. 117 (1993), no. 1, 215–242.