A nonlinear Schrödinger equation with fractional noise
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- by Aurélien Deya, Nicolas Schaeffer and Laurent Thomann PDF
- Trans. Amer. Math. Soc. 374 (2021), 4375-4422 Request permission
Abstract:
We study a stochastic Schrödinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension $d\leq 3$. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schrödinger group, which allows us to cope with the strong irregularity of the solution.References
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Additional Information
- Aurélien Deya
- Affiliation: Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
- ORCID: 0000-0001-8786-815X
- Email: aurelien.deya@univ-lorraine.fr
- Nicolas Schaeffer
- Affiliation: Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
- Email: nicolas.schaeffer@univ-lorraine.fr
- Laurent Thomann
- Affiliation: Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
- MR Author ID: 794415
- Email: laurent.thomann@univ-lorraine.fr
- Received by editor(s): May 1, 2020
- Received by editor(s) in revised form: October 6, 2020, and November 1, 2020
- Published electronically: March 30, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 4375-4422
- MSC (2020): Primary 60H15, 35Q55, 60G22
- DOI: https://doi.org/10.1090/tran/8368
- MathSciNet review: 4251233