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 L²(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.