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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
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References:

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