Some Problems in Stochastic Flows and Couplings of Diffusions

首席研究员将研究流形和随机流动上的扩散。扩散的研究主要针对耦合的各个方面及其应用。一个目标是发展一种方法来耦合由狄利克雷形式产生的扩散。这项工作的一个潜在应用是对Moser - Harnack不等式的概率证明。另一个目标是研究常用的镜面耦合在流形上的效率。这项工作的一个目标是获得几何对光谱底部影响的潜在新见解。第三个问题是确定以归一化黎曼体积为平衡分布的扩散收敛到平衡的速度最快。流动中的主要问题集中在随机流动下物体的分散速率。本研究集中于粒子在称为流形的几何结构上的扩散。它起源于布朗运动理论，并利用了众所周知的扩散和热在物体（流形）中的传播之间的联系。一部分研究流形上温度分布的变化率，另一部分研究稳态温度分布的收敛速率。这些可以用扩散理论来研究，并且可以确定流形几何形状对这些量的影响。研究随机流的扩散本质上是确定如果所有的油粒子都在进行扩散或布朗运动，浮油扩散的速度有多快。这项工作可应用于污染控制问题。

The principal investigator will study diffusions on manifolds and stochastic flows. The work on diffusions is aimed at various aspects of coupling and their applications. One goal is to develop a method for coupling a diffusion which arises from a Dirichlet form. A potential application of this work is a probabilistic proof of the Moser Harnack inequality. Another goal is to study the efficiency of the commonly used mirror coupling on manifolds. A goal of this work is to gain potential new insights of the influence of geometry on the bottom of the spectrum. A third problem is to determine which diffusions with the normalized Riemannian volume as equilibrium distribution converge to equilibrium the fastest. The principal problems in flows focus on rates of dispersion of bodies under stochastic flows. This research concentrates on the diffusion of particles on geometric structures called manifolds. It arises from the theory of Brownian motion and exploits well known connections between diffusions and the spread of heat in bodies (manifolds). Part of the research deals with rates of change of temperature distributions on manifolds while another deals with the rate of convergence to a steady state temperature distribution. These can be studied by diffusion theory and the influence of the geometry of the manifold on these quantities can be determined. The work on dispersion of stochastic flows is essentially the determination of how fast an oil slick will spread if all the oil particles are undergoing diffusive or Brownian motion. Applications of this work can be made to pollution control problems.