A Stochastic Problem in Physics

The world is global and stochastic and physical laws are local and deterministic, Thus the problems discussed at this Summer School are the very fabric of physics. But physics asks some questions which go beyond the territory which has been explored here. I shall present one of them, show how far physicists have gone toward its solution and mention an important problem of current interest.

Probability theory begins with a probability space (Ω,ℱ,P). The careful definition of the g-field ℱ of subsets of Ω and of the probability measure p has given us many powerful theorems. It is also possible, and often preferable in physics, to define P as a promeasure,

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namely as a projective family of bounded measures defined on the system of finite dimensional spaces Q known as the projective system of Ω. We thus start from (Ω,Q,P) rather than (Ω,ℱ,P). This is excellent for statistical mechanics. Unfortunately, in quantum mechanics, we have to deal with families of unbounded measures on the projective system Q of Ω. And this is the'key issue in the study of Feynman path integrals.