A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs |
| 2. | Creator | Author's name, affiliation, country | Martin Hairer; University of Warwick; United Kingdom |
| 2. | Creator | Author's name, affiliation, country | Jonathan C Mattingly; Duke University; United States |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | Hypoellipticity; Hörmander condition; stochastic PDE |
| 3. | Subject | Subject classification | 60H15; 60H07; 35H10 |
| 4. | Description | 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. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | EPSRC; NSF; Sloan foundation |
| 7. | Date | (YYYY-MM-DD) | 2011-03-30 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/875 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v16-875 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 16 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
| 15. | Rights | Copyright and permissions | The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available. Summary of the Creative Commons Attribution License You are free
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