Random mass flow through random potential

Anderson哈密顿量及其现象学最初被引入固体物理学中，用来模拟呈现无序诱导的金属-绝缘体转变的非晶态材料和合金，现在已经成为随机背景导致的效应相关性的典范例子。由该算符控制的随机运动的一个流行模型是抛物线安德森模型(PAM)，即由安德森哈密顿量描述的具有随机源和随机汇的离散热方程。其解的一个特征性质是在大时限内出现间歇性峰值。他们使用路径积分方法和大偏差理论确定了概率文献中广泛研究的热力学观测数据。在更概率的语言中，通过随机势的随机质量流集中在彼此相距很远的小岛屿上。自20世纪90年代初以来，人们对PAM进行了大量的数学研究。对于一些非常特殊的势分布，近年来已经证明了这些岛屿的间歇图和许多详细的性质。然而，对PAM所描述的质量流随时间演变的分析仍处于初级阶段。此外，近年来获得的关于PAM的知识还没有被用于回答更复杂的模型的雄心勃勃的问题，比如具有随机扩散速率的PAM，或者由稳定运动或非对称随机漫步取代的扩散率。此外，对弱无序(相当于扩散系数的加速)对PAM和Anderson哈密顿主本征值的影响的理解，到目前为止仅限于一些非常特殊的情况。在该项目中，所有这些问题都被调查了，并揭示了在特殊情况下所知的潜在影响的一般原理。首先，我们证明了PAM的质量只集中在一个岛上，并研究了这个岛的位置的老化和亚稳性，以及关于随机质量流的时间演化的其他详细问题。其次，我们研究了弱无序对PAM的影响以及对广义主本征值的影响。最后，我们用随机电导之间的随机游动或稳定随机游动或非对称随机游动的生成元取代了Anderson哈密顿量的扩散部分，即拉普拉斯算子，研究了PAM。

Originally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson Hamiltonian and its phenomenology has become a paradigmatic example for the relevance of effects resulting from a random background. A popular model for the random motion governed by this operator is the parabolic Anderson model (PAM), the discrete heat equation with random sources and sinks modeled by the Anderson Hamiltonian. A characteristic property of its solutions is the occurrence of intermittency peaks in the large time limit. They determine the thermodynamic observables extensively studied in the probabilistic literature using path integral methods and the theory of large deviations. In more probabilistic language, the random mass flow through the random potential concentrates on small islands, which are separated far from each other. Since the beginning of the 1990s, a lot of mathematical research has been carried out on the PAM. For some very special potential distributions, the intermittency picture and many detailed properties of the islands have been proved in recent years. However, the analysis of the time-evolution of the mass flow described by the PAM is still in its infancy. Furthermore, the knowledge about the PAM gained in recent years has not yet been applied to answer ambitious questions about more complex models, like the PAM with random diffusion rates or with the diffusivity replaced by a stable motion or by a non-symmetric random walk. Also the understanding of the influence of weak disorder (equivalently, acceleration of the diffusivity), both for the PAM and for the principal eigenvalue of the Anderson Hamiltonian, is so far restricted to some very special cases. In the project, all these questions are investigated, and the general principles underlying the effects, known in special cases, are brought to the surface. First, we prove that the mass of the PAM asymptotically concentrates on just one island, and ageing and metastability properties of the location of this island and other detailed questions about the time-evolution of the random mass flow are studied. Second, we investigate the influence of weak disorder on the PAM and on the principal eigenvalue in broad generality. Finally, we study the PAM after replacing the diffusive part of the Anderson Hamiltonian, the Laplace operator, by the generator of a random walk among random conductances or a stable random walk or a non-symmetric random walk.