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Uniform Large Deviations for the Nonlinear Schrödinger Equation with Multiplicative Noise

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  • Eric Gautier

    (Crest)

Abstract

Uniform large deviations for the laws of the paths of the solutionsof the stochastic nonlinear Schr¨odinger equation when the noise converges tozero are presented. The noise is a real multiplicative Gaussian noise. It iswhite in time and colored in space. The path space considered allows blow-upand is endowed with a topology analogue to a projective limit topology. Thusa large variety of large deviation principle may be deduced by contraction. Asa consequence, asymptotics of the tails of the law of the blow-up time whenthe noise converges to zero are obtained.

Suggested Citation

  • Eric Gautier, 2004. "Uniform Large Deviations for the Nonlinear Schrödinger Equation with Multiplicative Noise," Working Papers 2004-42, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2004-42
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    References listed on IDEAS

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    1. Rasmussen, K.Ø. & Gaididei, Yu.B. & Bang, O. & Christiansen, P.L., 1996. "Nonlinear and stochastic modelling of energy transfer in Scheibe aggregates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 40(3), pages 339-358.
    2. Arnaud Debussche & Eric Gautier, 2005. "Small Noise Asymptotic of the Timing Jitter in Soliton Transmission," Working Papers 2005-20, Center for Research in Economics and Statistics.
    3. Ledoux, M. & Qian, Z. & Zhang, T., 2002. "Large deviations and support theorem for diffusion processes via rough paths," Stochastic Processes and their Applications, Elsevier, vol. 102(2), pages 265-283, December.
    4. Chenal, Fabien & Millet, Annie, 1997. "Uniform large deviations for parabolic SPDEs and applications," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 161-186, December.
    5. Cardon-Weber, Caroline, 1999. "Large deviations for a Burgers'-type SPDE," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 53-70, November.
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    Cited by:

    1. Salins, M., 2021. "Systems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial data," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 159-194.
    2. Meng, Lixin & Li, Jingyu & Tao, Jian, 2017. "Global energy solutions to a stochastic Schrödinger–Poisson system with multiplicative noise in two dimensions," Applied Mathematics and Computation, Elsevier, vol. 300(C), pages 40-59.
    3. Chen, Chuchu & Dang, Tonghe & Hong, Jialin, 2024. "An adaptive time-stepping fully discrete scheme for stochastic NLS equation: Strong convergence and numerical asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 173(C).

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