Some Problems in Stochastic Flows and Diffusions

本研究涵盖了随机流动和扩散耦合两个领域的问题。后一个领域的主要问题包括莫泽哈纳克不等式的概率证明和流形上最佳耦合的搜索。在前者领域，问题涉及布朗流和湍流的定性和定量行为，特别是它们的色散性质。这组耦合问题的一般设置是流形（例如曲面）上稳态温度分布的行为以及收敛到该稳态的速率。研究随机流的范围相当广泛。它目前关注的是浮油或高温体在海洋表面扩散的性质。最终目标是应用以前和当前工作中的一些想法来解决有关太阳磁场能量分形性质的问题。

This research encompasses problems in the two areas of stochastic flows and coupling of diffusions. The principal problems in the latter area involve a probabilistic proof of Moser's Harnack inequality and a search for the best coupling on manifolds. In the former area, the problems concern the qualitative and quantitative behavior of Brownian and turbulent flows, especially their dispersion properties.The general setting of the set of problems on coupling is the behavior of steady state temperature distributions on manifolds (surfaces for example) and the rate of convergence to this steady state. The scope of the work on stochastic flows is quite broad. It presently concerns the nature of dispersion of oil slicks or high temperature bodies on the ocean's surface. The ultimate goal is to apply some of the ideas from previous and current work to solve problems about the fractal nature of the energy of the magnetic field on the sun.