Logarithmic Sobolev inequalities and spectral concentration for the cubic Schrödinger equation

IF 0.8 4区数学 Q3 MATHEMATICS, APPLIED

Stochastics-An International Journal of Probability and Stochastic Processes Pub Date : 2013-08-16 DOI:10.1080/17442508.2014.882924

G. Blower, C. Brett, I. Doust

引用次数: 4

Abstract

The nonlinear Schrödinger equation , , arises from a Hamiltonian on infinite-dimensional phase space . For , Bourgain (Comm. Math. Phys. 166 (1994), 1–26) has shown that there exists a Gibbs measure on balls in phase space such that the Cauchy problem for is well posed on the support of , and that is invariant under the flow. This paper shows that satisfies a logarithmic Sobolev inequality (LSI) for the focusing case and on for all N>0; also satisfies a restricted LSI for on compact subsets of determined by Hölder norms. Hence for p = 4, the spectral data of the periodic Dirac operator in with random potential subject to are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of Korteweg–de Vries.

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对数Sobolev不等式和三次Schrödinger方程的谱浓度

非线性Schrödinger方程由无限维相空间上的哈密顿量产生。For, Bourgain (Comm. Math)。Phys. 166(1994)， 1-26)证明了在相空间中球上存在一个吉布斯测度，使得柯西问题在的支持下被很好地提出，并且在流动下是不变的。本文证明了在聚焦情况下满足对数Sobolev不等式(LSI)，在所有N>0的情况下满足对数Sobolev不等式;也满足由Hölder规范确定的紧子集的限制LSI。因此，当p = 4时，受随机势约束的周期狄拉克算子的谱数据集中在其均值附近。本文对希尔方程的谱数据作了类似的结论，当势是随机的，并服从Korteweg-de Vries的Gibbs测度。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Stochastics-An International Journal of Probability and Stochastic Processes MATHEMATICS, APPLIED-STATISTICS & PROBABILITY

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Stochastics: An International Journal of Probability and Stochastic Processes is a world-leading journal publishing research concerned with stochastic processes and their applications in the modelling, analysis and optimization of stochastic systems, i.e. processes characterized both by temporal or spatial evolution and by the presence of random effects. Articles are published dealing with all aspects of stochastic systems analysis, characterization problems, stochastic modelling and identification, optimization, filtering and control and with related questions in the theory of stochastic processes. The journal also solicits papers dealing with significant applications of stochastic process theory to problems in engineering systems, the physical and life sciences, economics and other areas. Proposals for special issues in cutting-edge areas are welcome and should be directed to the Editor-in-Chief who will review accordingly. In recent years there has been a growing interaction between current research in probability theory and problems in stochastic systems. The objective of Stochastics is to encourage this trend, promoting an awareness of the latest theoretical developments on the one hand and of mathematical problems arising in applications on the other.