Mathematical Sciences: RUI: Construction of Quasi-Invariant Measures on Infinite Dimensional Vector Spaces

主要研究人员将研究在无限维希尔伯特空间上构造一类拟(平移)不变概率测度的问题。所研究的措施是由椭圆型Hilbert值随机微分方程诱导的措施。前人已经证明了这类测度是可微的，并建立了测度可微性与拟不变性之间的关系。他现在打算研究假设，使他能够将这些结果结合起来，朝着预期的目标前进。主要的困难是对诱导措施的某些规律性的验证。他打算通过扩散的时间反转方法来解决这个问题。主要研究人员正在研究随机微分方程(SDE)的某些性质，这是一种源于受随机波动影响的物理系统的动力学的方程。(这种系统的一个典型例子是飞机的位置，它的运动受到不稳定的大气条件的影响。)基于他以前在这个领域的一些工作，作者打算研究SDE解的所谓的准不变性。准不变性问题与量子场论有关，因此也是数学物理学家感兴趣的问题。

The principal investigator will address the problem of constructing a class of quasi-(translation) invariant probability measures on an infinite-dimensional Hilbert space. The measures to be studied are those induced by Hilbert-valued stochastic differential equations of elliptic type. The proposer has previously shown such measures to be differentiable, and has also established a relationship between the properties of measure differentiability and quasi-invariance. He now intends to investigate hypotheses which will allow him to combine these results, towards the desired end. The main difficulty is the verification of certain regularity properties of the induced measures. He intends to address this problem via a method of time-reversal for diffusions. The principal investigator is studying certain properties of stochastic differential equations (sde's), a type of equation which arises from the dynamics of a physical system which is subject to random fluctuations. (A typical example of such a system is the position of an aircraft, whose motion is influenced by erratic atmospheric conditions.) Building on some of his previous work in this area, the reviewer intends to study the so called quasi-invariance properties of solutions to sde's. The quasi-invariance problem is related to quantum field theory and as such is also of interest to mathematical physicists.