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Annals of PDE

, 4:16 | Cite as

Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

  • Nathan E. Glatt-Holtz
  • David P. Herzog
  • Jonathan C. Mattingly
Manuscript
  • 63 Downloads

Abstract

We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.

Keywords

Geometric control theory Stochastic partial differential equations (SPDEs) Degenerate stochastic forcing and hypoellipticity Malliavin calculus Fluid turbulence 

Mathematics Subject Classification

35Q35 35R60 60H15 60H07 76F70 

Notes

Acknowledgements

This work was initiated when the three authors were research members at the Mathematical Science Research Institute (MSRI) under the “New Challenges in PDE: Deterministic Dynamics and Randomness in High Infinite Dimensional Systems” program held in the Fall 2015. We are also grateful for the hospitality and travel support provided by the mathematics departments at Iowa State University and Tulane University which hosted a number of research visits that facilitated the completion of this work. We would like to warmly thank Juraj Földes, Susan Friedlander and Vlad Vicol for numerous helpful discussions and encouraging feedback on this work. Our efforts were supported in part through grants DMS-1313272 (NEGH), DMS-1612898 (DPH) and DMS-1613337 (JCM) from the National Science Foundation.

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Authors and Affiliations

  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA
  2. 2.Department of MathematicsIowa State UniversityAmesUSA
  3. 3.Department of MathematicsDuke UniversityDurhamUSA

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