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Journal of Nonlinear Science

, Volume 24, Issue 3, pp 383–409 | Cite as

Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

  • Viorel Barbu
  • Michael Röckner
  • Deng Zhang
Article

Abstract

We prove well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise, including the nonconservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower-order terms and treat the resulting equation by a fixed point argument based on generalizations of Strichartz estimates proved by Marzuola et al. (J Funct Anal 255(6):1479–1553, 2008). This approach makes it possible to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range \([1, 1 + 4/d)\) of admissible exponents in the nonlinear part (where \(d\) is the dimension of the underlying Euclidean space), i.e., in exactly the same range as in the deterministic case.

Keywords

Stochastic Wiener noise Strichartz estimate 

Mathematics Subject Classification

35Q41 60H15 35R60 

Notes

Acknowledgments

The authors are indebted to Daniel Tataru for fruitful discussions and suggestions regarding the proof of Lemma 4.1 Tataru.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Octav Mayer Institute of MathematicsRomanian AcademyIasiRomania
  2. 2.Al.I. Cuza UniversityIasiRomania
  3. 3.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  4. 4.Institute of Applied Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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