从非布朗泛函到分数阶薛定谔方程
From non-Brownian Functionals to a Fractional Schrödinger Equation
摘要：
本文导出了粒子反常扩散路径泛函分布的向后和向前分数阶薛定谔型方程。Friedrich及其同事[PRL {\bf 96}，230601（2006）]引入的分数实质导数为手头的问题提供了正确的分数框架。在正常扩散的极限下，我们恢复了布朗泛函的Feynman-Kac处理。作为应用，我们计算了半空间中占据时间的分布，并展示了反常泛函的统计与弱遍历破缺的关系。
英文摘要：
We derive backward and forward fractional Schrödinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.
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