Mathematical Sciences: Geometric Aspects of Random Motion

9626428克兰斯顿摘要首席研究员攻击了随机分析一般领域中的几个问题。其中一个问题是给出一种扩散耦合，它的简并生成元由满足霍曼德亚椭圆性条件的向量平方和给出。一个相关的问题是给出了Moser的Harnack不等式的耦合证明。主要研究人员还利用Gromov双曲空间上的布朗运动的耦合技术来估计调和函数的特征值和梯度。首席研究员还研究了相对论布朗运动。最后，研究人员正在得出更多关于依赖于过去的过程的行为的结果。目前提案中的工作属于随机分析的一般领域。这意味着对随机演变现象的分析。这一建议的耦合部分与表面温度分布的平稳性、收敛到稳定状态的速度以及这些对表面几何形状的依赖有关。有关其行为依赖于过去的过程的问题具有潜在的经济学应用(例如价格)以及生物学(趋化现象)。相对论布朗运动的工作在宇宙学中有潜在的应用，特别是在决定是否有可能观测黑洞方面。

9626428 Cranston ABSTRACT The principal investigator attacks several problems in the general area of Stochastic Analysis. One problem is to come up with a coupling for diffusions with degenerate generators given by sums of squares of vector which satisfy Hormander's hypoellipticity condition. A related problem is to give a coupling proof of Moser's Harnack inequality. The principal investigator also uses coupling techniques on Brownian motion on Gromov hyperbolic spaces to estimate eigenvalues and gradients of harmonic functions. The principal investigator also studies relativistic Brownian motion. Finally, the investigator is deriving more results on the behavior of processes with dependence on the past. The work in the current proposal falls into the general area of Stochastic Analysis. This means the analysis of randomly evolving phenomena. The portions of this proposal on coupling are related to the smoothness of temperature distributions on surfaces, their rate of convergence to a steady state and the dependence of these on the geometry of the surface. The problems on processes whose behavior depends on the past have potential applications to economics (prices e.g.) and to biology (chemotactic phenomena.) The work on relativistic Brownian motion can have potential applications to cosmology, especially in deciding whether it is possible to observe black holes.