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

首席研究员将解决以下问题： 一类拟平移不变概率的构造 无限维希尔伯特空间上的测度。 的措施 要研究的是由Hilbert值随机 椭圆型微分方程 提议者有 以前表明这种措施是可区分的，也 建立了测度性质之间的关系 可微性和拟不变性。 他现在打算 研究假设，使他能够将这些联合收割机 结果，朝着理想的方向。 主要的困难是 验证诱导的某些规律性性质 措施 他打算通过一种方法来解决这个问题， 扩散的时间反演。 首席研究员正在研究 随机微分方程（sde），一类方程 它源于一个物理系统的动力学， 受到随机波动的影响。 （一个典型的例子， 系统是飞机的位置，其运动受到影响 不稳定的大气条件）。 建立在他的一些 以前的工作在这方面，评论家打算研究的，所以 称为半微分方程解的准不变性质。 的 准不变性问题与量子场论有关， 这也是数学物理学家感兴趣的。

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.