标题： 一维线性振子系统中Wigner变换的长时间大尺度极限 显示英文标题

标题： Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

摘要： 我们考虑对应于一维整数格点上的离散波动方程的波函数的Wigner变换$W_\eps(t,x,k)$在长时间、大尺度下的行为，带有弱乘性噪声。 这样的波函数对应于该模型已被Basile、Bernardin和Olla引入，以描述一个相互作用的线性振子系统，其中存在局部保持动能和动量的弱噪声。 Wigner变换的动能极限已在Basile、Olla和Spohn中得到证明。 在本文中，我们证明在未固定的情况下存在$\gamma_0>0$，使得对于任何$\gamma\in(0,\gamma_0]$，当$\eps\ll1$时，$W_\eps(t/\eps^{1+3/2\gamma},x/\eps^{\gamma},k)$的弱极限满足一维分数热方程$\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$，其中适当的系数为$\hat c>0$。 在固定情况下，可以对$W_\eps(t/\eps^{1+2\gamma},x/\eps^{\gamma},k)$得出类似的结果，但此时极限满足普通的热方程。 显示英文摘要

摘要： We consider the long time, large scale behavior of the Wigner transform $W_\eps(t,x,k)$ of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. %Such a wave function corresponds to This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $\gamma_0>0$ such that for any $\gamma\in(0,\gamma_0]$ the weak limit of $W_\eps(t/\eps^{1+3/2\gamma},x/\eps^{\gamma},k)$, as $\eps\ll1$, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ for an appropriate coefficient $\hat c>0$. In the pinned case an analogous result can be claimed for $W_\eps(t/\eps^{1+2\gamma},x/\eps^{\gamma},k)$ but the limit satisfies then the usual heat equation.