Localisation and ageing in the parabolic Anderson model with Weibull potential
Localisation and ageing in the parabolic Anderson model with Weibull potential
复制标题
具有威布尔势的抛物线安德森模型中的局域化和老化
DOI：
10.1214/13-aop882
复制
发表时间：
2012
影响因子：
2.3
通讯作者：
Aleksander Twarowski
中科院分区：
文献类型：
--
作者：
N. Sidorova;Aleksander Twarowski
抛物型安德森模型是具有随机势$\xi$的整数格点上热方程的柯西问题。我们考虑的情况下，当$\{\xi（z）：z\in\mathbb{Z}^d\}$是一个收集的独立同分布的随机变量的威布尔分布的参数为$0<\gamma<2$，我们假设的解决方案是最初定位在原点。我们证明，随着时间的推移到无穷大，解决方案完全本地化在只有一个点具有很高的概率，我们确定的本地化网站的渐近行为。我们还表明，当溶液从一个站点重新定位到另一个站点时，时间之间的间隔随时间线性增加，这种现象称为老化。
The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case when $\{\xi(z):z\in\mathbb{Z}^d\}$ is a collection of independent identically distributed random variables with Weibull distribution with parameter $0<\gamma<2$, and we assume that the solution is initially localised in the origin. We prove that, as time goes to infinity, the solution completely localises at just one point with high probability, and we identify the asymptotic behaviour of the localisation site. We also show that the intervals between the times when the solution relocalises from one site to another increase linearly over time, a phenomenon known as ageing.