Mathematical Sciences: Diffusion Processes and Related Topics

9625782 Stroock摘要 主要研究者打算继续他的研究，以各种问题的路径空间上的集成的应用。 在过去，他帮助开发的技术，导致推导公式的解决方案，某些偏微分方程的积分方面超过一个适当的路径空间。 然后，他使用这些公式，以获得相当精细的估计的解决方案表示。 最近，他一直试图使用相关的方法来深入了解微分几何问题建议最近的物理考虑。 特别是，他希望学习如何构建和分析扩散空间的循环在黎曼流形。 在牛顿的世界图景中，物理对象被认为遵循确定的路径，物理学的目标是预测特定对象将遵循的路径。 在量子力学的图景中，物理对象不再被认为是如此受约束。 相反，它们被认为是受物理定律支配的，这些定律最多只能预测它们沿着特定路径运动的概率，牛顿和量子图景被这样一个事实所调和，即一个大质量物体以压倒性的概率沿着一条接近牛顿物理学预测的路径运动。 另一方面，当处理非常小的物体时，牛顿的预测可能会非常遥远，人们只能通过对路径空间进行平均来进行预测。 因此，了解如何对路径求平均值变得非常重要。 在各种路径空间上求平均的数学理论一直是本世纪数学的中心主题之一。 它的应用范围很广，并不局限于物理学。 事实上，在过去20年中，非常复杂的路径空间模型是各种商品和金融工具定价的基础。 路径空间的平均化是这个项目中提出的研究的中心主题。

9625782 Stroock ABSTRACT The principal investigator intends to continue his research into the applications of integration on path spaces to various problems. In the past, he helped develop techniques which lead to the derivation of formulae for solutions to certain partial differential equations in terms of integrals over an appropriate path space. He then used these formulae to obtain quite refined estimates on the solutions represented. More recently, he has been trying to use related methods to gain insight into differential geometric problems suggested by recent physical considerations. In particular, he hopes to learn how to construct and analyze diffusions on spaces of loops in a Riemannian manifold. In the Newtonian picture of the world, physical objects were thought to follow definite paths, and the goal of physics was to predict the path that a particular object would follow. In the quantum mechanical picture, physical objects are no longer thought to be so constrained. Instead, they are thought to governed by physical laws which can, at best, predict the probability of their following a particular path. The Newtonian and quantum picture are reconciled by the fact that, with overwhelming probability, a massive object will follow a path which is close to the one predicted by Newtonian physics. On the other hand, when dealing with very small objects, the Newtonian prediction can be quite far off, and one can only make predictions by averaging over spaces of paths. For this reason, it becomes important to understand how to average over paths. The mathematical theory of averaging over various path spaces has been one of the central themes of mathematics in this century. Its applications are numerous and are not restricted to physics. Indeed, over the past two decades, very sophisticated path-space models are the basis on which commodities and financial instruments of all sorts have been priced. Averaging over path spaces is the central theme of the research proposed in this project.