Analysis and Probability over Infinite Extensions of a Local Field

摘要

We consider an infinite extension K of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. K is equipped with an inductive limit topology; its conjugate K-bar is a completion of K with respect to a topology given by certain explicitly written seminorms. We construct and study a Gaussian measure, a Fourier transform, a fractional differentiation operator and a cadlag Markov process on K-bar. If we deal with Galois extensions then all these objects are Galois-invariant.