A heat equation on some adic completions of ℚ and ultrametric analysis

Abstract

For each finite set S of prime numbers there exists a unique completion ℚ S of ℚ, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D α and to study an abstract heat equation on the Hilbert space L 2(ℚ S ). The fundamental solution of this equation is a normal transition function of a Markov process on ℚ S . The techniques developed provides a general framework for these kind of problems on different ultrametric groups.