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

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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.