A lift of spatially inhomogeneous Markov process to extensions of the field of -adic numbers

A Markov process on a local field which can be projected to a Markov process on a smaller local field is regarded as a lift of the one on the smaller field. The first part of this article is concerned with a Markov process on a local field which is obtained as the one projected from a larger field by means of the algebraic trace. Since the explicit expression of the transition probability plays important roles in a study of Markov processes on local fields, the second part is devoted to finding an explicit expression for the Markov process.