  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 $\bar{K}$ 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
$\bar{K}$. If we deal with Galois extensions then all these objects are
Galois-invariant.
