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

• Published in: P-adic numbers, ultrametric analysis, and applications Vol. 9; no. 3; pp. 165 - 182
• Main Authors: Aguilar-Arteaga, V. A., Cruz-López, M., Estala-Arias, S.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-07-2017; Springer Nature B.V
• Subjects: Algebra; Hilbert space; Markov analysis; Markov processes; Mathematics; Mathematics and Statistics; Prime numbers; Topology; ultrametrics; Markov processes; pseudodifferential equations; heat kernels; adic completions
• ISSN: 2070-0466; 2070-0474
• DOI: 10.1134/S2070046617030013

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.