Hausdorff measure for a stable-like process over an infinite extension of a local field

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) over bar is a completion of K with respect to a topology given by certain explicitly written seminorms. The semigroup of measures, which defines a stable-like process X(t) on (K) over bar, is concentrated on a compact subgroup S subset of (K) over bar. We study properties of the process X-S(t), a part of X(t) in S. It is shown that the Hausdorff and packing dimensions of the image of an interval equal 0 almost surely. In the case of tamely ramified extensions a correct Hausdorff measure for this set is found.