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**Rights statement:**The final, definitive version of this article has been published in the Journal, Stochastics, 86 (6), 2014, © Informa PlcSubmitted manuscript, 178 KB, PDF document

Research output: Contribution to journal › Journal article › peer-review

Published

<mark>Journal publication date</mark> | 2014 |
---|---|

<mark>Journal</mark> | Stochastics |

Issue number | 6 |

Volume | 86 |

Number of pages | 12 |

Pages (from-to) | 870-881 |

Publication Status | Published |

Early online date | 19/03/14 |

<mark>Original language</mark> | English |

The nonlinear Schr\"odinger equation $\NLSE(p, \beta)$,

$-iu_t=-u_{xx}+\beta \vert u\vert^{p-2} u=0$,

arises from a Hamiltonian on infinite-dimensional phase space

$\Lp^2(\mT)$. For $p\leq 6$, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure $\mu^{\beta}_N$ on balls $\Omega_N=

\{ \phi\in \Lp^2(\mT) \,:\, \Vert \phi \Vert^2_{\Lp^2} \leq N\}$

in phase space such that the Cauchy problem for $\NLSE(p,\beta)$ is well posed on the support of $\mu^{\beta}_N$, and that $\mu^{\beta}_N$ is invariant under

the flow.

This paper shows that $\mu^{\beta}_N$ satisfies a logarithmic Sobolev inequality for the focussing case $\beta <0$ and $2\leq p\leq 4$

on $\Omega_N$ for all $N>0$; also $\mu^{\beta}$ satisfies a restricted LSI for $4\leq p\leq 6$ on compact subsets of $\Omega_N$ determined by H\"older

norms. Hence for $p=4$, the spectral data of the periodic Dirac operator

in $\Lp^2(\mT; \mC^2)$ with random potential $\phi$ subject to $\mu^{\beta}_N$ 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 KdV.

$-iu_t=-u_{xx}+\beta \vert u\vert^{p-2} u=0$,

arises from a Hamiltonian on infinite-dimensional phase space

$\Lp^2(\mT)$. For $p\leq 6$, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure $\mu^{\beta}_N$ on balls $\Omega_N=

\{ \phi\in \Lp^2(\mT) \,:\, \Vert \phi \Vert^2_{\Lp^2} \leq N\}$

in phase space such that the Cauchy problem for $\NLSE(p,\beta)$ is well posed on the support of $\mu^{\beta}_N$, and that $\mu^{\beta}_N$ is invariant under

the flow.

This paper shows that $\mu^{\beta}_N$ satisfies a logarithmic Sobolev inequality for the focussing case $\beta <0$ and $2\leq p\leq 4$

on $\Omega_N$ for all $N>0$; also $\mu^{\beta}$ satisfies a restricted LSI for $4\leq p\leq 6$ on compact subsets of $\Omega_N$ determined by H\"older

norms. Hence for $p=4$, the spectral data of the periodic Dirac operator

in $\Lp^2(\mT; \mC^2)$ with random potential $\phi$ subject to $\mu^{\beta}_N$ 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 KdV.

Submitted for publication, and posted on ArXiv 1308:3649
The final, definitive version of this article has been published in the Journal,
Stochastics, 86 (6), 2014, © Informa Plc