## Markovianity and ergodicity for a surface growth PDE.(English)Zbl 1184.60024

The authors consider the following model arising in the theory of growth of surfaces: $\dot h = -h_{xxxx}-h_{xx} + (h_x^2)_{xx} + \eta,$ on $$[0,L]$$, with periodic boundary conditions, and $$\eta$$ is a space-time white noise. This type of boundary conditions (either Dirichlet or von Neumann) turns out to be essential to derive some convenient a priori estimates. On the other hand, the main problem in the analysis of this model is the lack of uniqueness of weak solutions.
First, the authors prove that there exists a weak solution having the Markov property. The method in the proof is based on the definition of weak martingale solutions and uses some ideas of [F. Flandoli and M. Romito, Probab. Theory Relat. Fields 140, No. 3–4, 407–458 (2008; Zbl 1133.76016)], where the latter deals with the 3D stochastic Navier-Stokes equation. Secondly, the strong Feller property of the corresponding transition semigroup is established. Finally, the authors study long time behaviour properties of the model. Namely, they prove that any Markov solution has a unique invariant measure whose support covers the whole state space. This result is closely related to the considerations in [D. Blömker and M. Hairer, Stochastic Anal. Appl. 22, No. 4, 903–922 (2004; Zbl 1057.60060)].
As explained in the introduction of the paper, the main mathematical interests of the model are the following: first, in comparison with Navier-Stokes equations, the {natural space} for the Markov dynamics has been completely determined and, secondly, the analysis of the energy inequality in the space-time white noise setting.

### MSC:

 60H30 Applications of stochastic analysis (to PDEs, etc.) 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35Q99 Partial differential equations of mathematical physics and other areas of application 35R60 PDEs with randomness, stochastic partial differential equations 82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics

### Citations:

Zbl 1133.76016; Zbl 1057.60060
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### References:

 [1] Barabási, A.-L. and Stanley, H. E. (1995). Fractal Concepts in Surface Growth . Cambridge Univ. Press, Cambridge. · Zbl 0838.58023 [2] Blömker, D. (2005). Nonhomogeneous noise and Q -Wiener processes on bounded domains. Stochastic Anal. Appl. 23 255-273. · Zbl 1068.60058 [3] Blömker, D. and Gugg, C. (2004). Thin-film-growth-models: On local solutions. In Recent Developments in Stochastic Analysis and Related Topics 66-77. World Scientific, Singapure. · Zbl 1080.60063 [4] Blömker, D. and Gugg, C. (2002). On the existence of solutions for amorphous molecular beam epitaxy. Nonlinear Anal. Real World Appl. 3 61-73. · Zbl 1027.60063 [5] Blömker, D., Gugg, C. and Raible, M. (2002). Thin-film-growth models: Roughness and correlation functions. European J. Appl. Math. 13 385-402. · Zbl 1020.82014 [6] Blömker, D. and Hairer, M. (2004). Stationary solutions for a model of amorphous thin-film growth. Stochastic Anal. Appl. 22 903-922. · Zbl 1057.60060 [7] Castro, M., Cuerno, R., Vázquez, L. and Gago, R. (2005). Self-organized ordering of nanostructures produced by ion-beam sputtering. Phys. Rev. Lett. 94 016102. [8] Collet, P., Eckmann, J.-P., Epstein, H. and Stubbe, J. (1993). A global attracting set for the Kuramoto-Sivashinsky equation. Comm. Math. Phys. 152 203-214. · Zbl 0777.35073 [9] Cuerno, R. and Barabási A.-L. (1995). Dynamic scaling of ion-sputtered surfaces. Phys. Rev. Lett. 74 4746-4749. [10] Da Prato, G. and Debussche, A. (2003). Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures Appl. (9) 82 877-947. · Zbl 1109.60047 [11] Da Prato, G. and Debussche, A. (2007). m -dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise. Potential Anal. 26 31-55. · Zbl 1119.35128 [12] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications 44 . Cambridge Univ. Press, Cambridge. · Zbl 0761.60052 [13] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229 . Cambridge Univ. Press, Cambridge. · Zbl 0849.60052 [14] Debussche, A. and Odasso, C. (2006). Markov solutions for the 3D stochastic Navier-Stokes equations with state dependent noise. J. Evol. Equ. 6 305-324. · Zbl 1110.35110 [15] Flandoli, F. (1997). Irreducibility of the 3-D stochastic Navier-Stokes equation. J. Funct. Anal. 149 160-177. · Zbl 0887.35171 [16] Flandoli, F. (2008). An introduction to 3D stochastic fluid dynamics. In Proceedings of the CIME Course on SPDE in Hydrodynamics: Recent Progress and Prospects. Lecture Notes in Mathematics 1942 51-150. Springer, Berlin. · Zbl 1426.76001 [17] Flandoli, F. and Romito, M. (2001). Statistically stationary solutions to the 3-D Navier-Stokes equation do not show singularities. Electron. J. Probab. 6 15 (electronic). · Zbl 0973.35148 [18] Flandoli, F. and Romito, M. (2006). Markov selections and their regularity for the three-dimensional stochastic Navier-Stokes equations. C. R. Math. Acad. Sci. Paris 343 47-50. · Zbl 1098.60059 [19] Flandoli, F. and Romito, M. (2008). Markov selections for the 3D stochastic Navier-Stokes equations. Probab. Theory Related Fields 140 407-458. · Zbl 1133.76016 [20] Flandoli, F. and Romito, M. (2007). Regularity of transition semigroups associated to a 3D stochastic Navier-Stokes equation. In Stochastic Differential Equations: Theory and Applications. Interdiscip. Math. Sci. 2 263-280. World Scientific, Singapure. · Zbl 1136.35112 [21] Halpin-Healy, T. and Zhang, Y. C. (1995). Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Physics Reports 254 215-414. [22] Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 . Springer, Berlin. · Zbl 0456.35001 [23] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes , 2nd ed. North-Holland Mathematical Library 24 . North-Holland Publishing Co., Amsterdam. · Zbl 0684.60040 [24] Krylov, N. V. (1973). The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes. Izv. Akad. Nauk SSSR Ser. Mat. 37 691-708. · Zbl 0295.60057 [25] Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications 16 . Birkhäuser, Basel. · Zbl 0816.35001 [26] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44 . Springer, New York. · Zbl 0516.47023 [27] Raible, M., Linz, S. J. and Hänggi, P. (1691). Amorphous thin film growth: Minimal deposition equation. Phys. Rev. E 62 1691-1705. [28] Raible, M., Mayr, S. G., Linz, S. J., Moske, M., Hänggi, P. and Samwer, K. (2000). Amorphous thin film growth: Theory compared with experiment. Europhys. Lett. 50 61-67. [29] Romito M. (2006). Existence of martingale and stationary suitable weak solutions for a stochastic Navier-Stokes system. · Zbl 1277.76012 [30] Siegert, M. and Plischke, M. (1994). Solid-on-solid models of molecular-beam epitaxy. Phys. Rev. E 50 917-931. [31] Stein, O. and Winkler, M. (2005). Amorphous molecular beam epitaxy: Global solutions and absorbing sets. European J. Appl. Math. 16 767-798. · Zbl 1120.35045 [32] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 233 . Springer, Berlin. · Zbl 0426.60069 [33] Temam, R. (1984). Navier-Stokes Equations , 3rd ed. Studies in Mathematics and its Applications 2 . North-Holland, Amsterdam. · Zbl 0568.35002 [34] Temam, R. (1988). Infinite-dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences 68 . Springer, New York. · Zbl 0662.35001
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