Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Éric Gautier

ESAIM: Probability and Statistics (2010)

  • Volume: 9, page 74-97
  • ISSN: 1292-8100

Abstract

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Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive Gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.

How to cite

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Gautier, Éric. "Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications." ESAIM: Probability and Statistics 9 (2010): 74-97. <http://eudml.org/doc/104342>.

@article{Gautier2010,
abstract = { Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive Gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given. },
author = {Gautier, Éric},
journal = {ESAIM: Probability and Statistics},
keywords = {Large deviations; stochastic partial differential equations; nonlinear Schrödinger equations; white noise; projective limit; support theorem; blow-up; solitary waves.; stochastic partial differential equations; nonlinear Schrödinger equations; white noise; blow-up; solitary waves},
language = {eng},
month = {3},
pages = {74-97},
publisher = {EDP Sciences},
title = {Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications},
url = {http://eudml.org/doc/104342},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Gautier, Éric
TI - Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 74
EP - 97
AB - Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive Gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.
LA - eng
KW - Large deviations; stochastic partial differential equations; nonlinear Schrödinger equations; white noise; projective limit; support theorem; blow-up; solitary waves.; stochastic partial differential equations; nonlinear Schrödinger equations; white noise; blow-up; solitary waves
UR - http://eudml.org/doc/104342
ER -

References

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  1. R. Azencott, Grandes déviations et applications, in École d'été de Probabilité de Saint-Flour, P.L. Hennequin Ed. Springer-Verlag, Berlin. Lect. Notes Math.774 (1980) 1–176.  
  2. A. Badrikian and S. Chevet, Mesures cylindriques, espaces de Wiener et fonctions aléatoires Gaussiennes. Springer-Verlag, Berlin. Lect. Notes Math.379 (1974).  
  3. Y.M. Berezansky, Z.G. Sheftel and G.F. Us, Functional Analysis, Vol. 1. Oper. Theor. Adv. Appl.85 (1997) 125–134.  
  4. G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems. Oxford University Press, Oxford. Oxford Lect. Ser. Math. Appl.15 (1998).  
  5. T. Cazenave, An Introduction to Nonlinear Schrödinger Equations. Instituto de Matématica-UFRJ Rio de Janeiro, Brazil. Textos de Métodos Matématicos26 (1993).  
  6. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press: Cambridge, England. Encyclopedia Math. Appl. (1992).  
  7. A. de Bouard and A. Debussche, The Stochastic Nonlinear Schrödinger Equation in H1. Stochastic Anal. Appl.21 (2003) 97–126.  
  8. A. de Bouard and A. Debussche, On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation. Probab. Theory Relat. Fields123 (2002) 76–96.  
  9. A. de Bouard and A. Debussche, Finite time blow-up in the additive supercritical nonlinear Schrödinger equation: the real noise case. Contemp. Math.301 (2002) 183–194.  
  10. A. Debussche and L. Di Menza, Numerical simulation of focusing stochastic nonlinear Schrödinger equations. Phys. D162 (2002) 131–154.  
  11. S.A. Derevyanko, S.K. Turitsyn and D.A. Yakusev, Non-gaussian statistics of an optical soliton in the presence of amplified spontaneaous emission. Optics Lett.28 (2003) 2097–2099.  
  12. J.D. Deuschel and D.W. Stroock, Large Deviations. Academic Press, New York. Pure Appl. Math. (1986).  
  13. A. Dembo and O. Zeitouni, Large deviation techniques and applications (2nd edition). Springer-Verlag, New York. Appl. Math.38 (1998).  
  14. P.D. Drummond and J.F. Corney, Quantum noise in optical fibers. II. Raman jitter in soliton communications. J. Opt. Soc. Am. B18 (2001) 153–161.  
  15. L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, Grad. Stud. in Math.119 (1998).  
  16. G.E. Falkovich, I. Kolokolov, V. Lebedev and S.K. Turitsyn, Statistics of soliton-bearing systems with additive noise. Phys. Rev. E63 (2001) 025601(R).  
  17. G. Falkovich, I. Kolokolov, V. Lebedev, V. Mezentsev and S.K. Turitsyn, Non-Gaussian error probability in optical soliton transmission. Physica D195 (2004) 1–28.  
  18. É. Gautier, Uniform large deviations for the nonlinear Schrödinger equation with multiplicative noise. Preprint IRMAR, Rennes (2004). Submitted for publication.  
  19. T. Kato, On Nonlinear Schrödinger Equation. Ann. Inst. H. Poincaré, Phys. Théor. 46 (1987) 113–129.  
  20. V. Konotop and L. Vázquez, Nonlinear random waves. World Scientific Publishing Co., Inc.: River Edge, New Jersey (1994).  
  21. R.O. Moore, G. Biondini and W.L. Kath, Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems. Optics Lett.28 (2003) 105–107.  
  22. C. Sulem and P.L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse. Springer-Verlag, New York, Appl. Math. Sci. (1999).  
  23. J.B. Walsh, An introduction to stochastic partial differential equations, in École d'été de Probabilité de Saint-Flour, P.L. Hennequin Ed. Springer-Verlag, Berlin, Lect. Notes Math.1180 (1986) 265–439.  

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