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.
Mots-clés : large deviations, stochastic partial differential equations, nonlinear Schrödinger equations, white noise, projective limit, support theorem, blow-up, solitary waves
@article{PS_2005__9__74_0,
author = {Gautier, \'Eric},
title = {Large deviations and support results for nonlinear {Schr\"odinger} equations with additive noise and applications},
journal = {ESAIM: Probability and Statistics},
pages = {74--97},
publisher = {EDP-Sciences},
volume = {9},
year = {2005},
doi = {10.1051/ps:2005005},
mrnumber = {2148961},
zbl = {1136.60346},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps:2005005/}
}
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 PY - 2005 SP - 74 EP - 97 VL - 9 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2005005/ DO - 10.1051/ps:2005005 LA - en ID - PS_2005__9__74_0 ER -
%0 Journal Article %A Gautier, Éric %T Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications %J ESAIM: Probability and Statistics %D 2005 %P 74-97 %V 9 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps:2005005/ %R 10.1051/ps:2005005 %G en %F PS_2005__9__74_0
Gautier, Éric. Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 74-97. doi : 10.1051/ps:2005005. http://archive.numdam.org/articles/10.1051/ps:2005005/
[1] , 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. | Zbl
[2] and, Mesures cylindriques, espaces de Wiener et fonctions aléatoires Gaussiennes. Springer-Verlag, Berlin. Lect. Notes Math. 379 (1974). | MR | Zbl
[3] , and, Functional Analysis, Vol. 1. Oper. Theor. Adv. Appl. 85 (1997) 125-134. | Zbl
[4] , and, One-dimensional Variational Problems. Oxford University Press, Oxford. Oxford Lect. Ser. Math. Appl. 15 (1998). | MR | Zbl
[5] , An Introduction to Nonlinear Schrödinger Equations. Instituto de Matématica-UFRJ Rio de Janeiro, Brazil. Textos de Métodos Matématicos 26 (1993).
[6] and, Stochastic Equations in Infinite Dimensions. Cambridge University Press: Cambridge, England. Encyclopedia Math. Appl. (1992). | MR | Zbl
[7] and, The Stochastic Nonlinear Schrödinger Equation in . Stochastic Anal. Appl. 21 (2003) 97-126. | Zbl
[8] and, On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation. Probab. Theory Relat. Fields 123 (2002) 76-96. | Zbl
[9] and, Finite time blow-up in the additive supercritical nonlinear Schrödinger equation: the real noise case. Contemp. Math. 301 (2002) 183-194. | Zbl
[10] and, Numerical simulation of focusing stochastic nonlinear Schrödinger equations. Phys. D 162 (2002) 131-154. | Zbl
[11] , and, Non-gaussian statistics of an optical soliton in the presence of amplified spontaneaous emission. Optics Lett. 28 (2003) 2097-2099.
[12] and, Large Deviations. Academic Press, New York. Pure Appl. Math. (1986). | MR | Zbl
[13] and, Large deviation techniques and applications (2nd edition). Springer-Verlag, New York. Appl. Math. 38 (1998). | MR | Zbl
[14] and, Quantum noise in optical fibers. II. Raman jitter in soliton communications. J. Opt. Soc. Am. B 18 (2001) 153-161.
[15] , Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, Grad. Stud. in Math. 119 (1998). | Zbl
[16] ,, and, Statistics of soliton-bearing systems with additive noise. Phys. Rev. E 63 (2001) 025601(R).
[17] ,,, and, Non-Gaussian error probability in optical soliton transmission. Physica D 195 (2004) 1-28. | Zbl
[18] , Uniform large deviations for the nonlinear Schrödinger equation with multiplicative noise. Preprint IRMAR, Rennes (2004). Submitted for publication. | MR | Zbl
[19] , On Nonlinear Schrödinger Equation. Ann. Inst. H. Poincaré, Phys. Théor. 46 (1987) 113-129. | Numdam | Zbl
[20] and, Nonlinear random waves. World Scientific Publishing Co., Inc.: River Edge, New Jersey (1994). | MR | Zbl
[21] , and, Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems. Optics Lett. 28 (2003) 105-107. | Zbl
[22] and, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse. Springer-Verlag, New York, Appl. Math. Sci. (1999). | MR | Zbl
[23] , 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. | Zbl
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