Abstract
We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.
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Acknowledgements
We thank the reviewers for their comments and suggestions.
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Open Access funding enabled and organized by Projekt DEAL. Rajeev Bhaskaran gratefully acknowledges financial support from the Science and Engineering Research Board (SERB), India – vide grant CRG/2019/002594. Stefan Tappe gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 444121509, and from the Deutsche Mathematiker-Vereinigung (DMV, German Mathematical Society) – Fachgruppe Stochastik.
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Bhaskaran, R., Tappe, S. Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10134-8
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DOI: https://doi.org/10.1007/s11118-024-10134-8
Keywords
- Stochastic partial differential equation
- Continuously embedded Hilbert spaces
- Invariant manifold
- Finite dimensional diffusion
- Multi-parameter group
- Hermite Sobolev space
- Translation invariant solution