Mathematical Sciences: Diffusion Processes and Related Topics

9625782 Stroock摘要首席研究员打算继续研究路空间上的积分在各种问题中的应用。在过去，他帮助开发了一些技术，这些技术导致了某些偏微分方程解的公式在适当的路径空间上的积分方面的推导。然后，他使用这些公式对所代表的解进行了相当精确的估计。最近，他一直在尝试使用相关的方法来洞察最近的物理考虑所提出的微分几何问题。特别是，他希望学习如何构造和分析黎曼流形中环空间上的扩散。在牛顿的世界图景中，物理对象被认为遵循确定的路径，而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限。相反，他们被认为受到物理定律的支配，这些定律充其量只能预测他们沿着特定路径前进的可能性。牛顿和量子图景是一致的，事实是，以压倒性的可能性，一个大质量物体将沿着一条接近牛顿物理学预测的路径运行。另一方面，当处理非常小的物体时，牛顿预测可能会相当遥远，人们只能通过对路径空间进行平均来进行预测。因此，了解如何在路径上求平均变得很重要。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。它的应用很多，而且并不局限于物理学。事实上，在过去20年里，非常复杂的路径空间模型是大宗商品和各种金融工具定价的基础。路径空间上的平均是本项目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.