Intermittency phenomena in random media

本文研究了空间分布随机系统的两种模型，即抛物型安德森模型（PAM）和Mullins-Sekerka演化模型（MSB）中的Ostwald熟化效应。间歇性表现在系统的高度不规则的演变中，这是由越来越稀疏的极端行为区域的出现所决定的。我们开发了新的谱方法来研究PAM在极端区域的解的几何形状，并在高度不规则的随机环境中处理随机游动的次扩散和老化。对于PAM，我们还研究了精细的动力学问题，如跳跃运输和处理随时间变化的随机电位和漂移的不稳定性。对于MSE，我们建议在整个空间上构建一个版本的模型，并特别考虑随机定位粒子的碰撞，屏蔽效应和系统的渐近自相似性来研究它。

We investigate the effect of intermittency in two models of spatially distributed random systems, the parabolic Anderson model (PAM) and the Mullins-Sekerka evolution (MSB) for Ostwald ripening. Intermittency manifests itself in a highly irregular evolution of the system determined by the appearance of more and more sparsely distributed regions of extreme behaviour. We develop new spectral methods to study the geometry of the solution to the PAM in extreme regions and to treat subdiffusivity and aging for random walks in highly irregular random environment. For the PAM, we also study refined dynamic questions like the hopping transport and deal with intermittency for time-dependent and drifted random potentials. For the MSE, we propose to construct a version of the model on the entire space and study it with particular consideration of the collisions of the randomly located particles, the screening effect, and asymptotic self-similarity of the system.