Parabolic Type Equations and Markov Stochastic Processes on Adeles

Abstract

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations.

36 pages

References in corpus (5)

• p-Adic Mathematical Physics
• Eternal Symmetree
• First Passage Time Distribution and Number of Returns for Ultrametric Random Walk
• Taibleson Operators, p-adic Parabolic Equations and Ultrametric Diffusion
• Wavelet analysis on adeles and pseudo-differential operators

Cited by in corpus (8)

• $p$-Adic Mathematical Physics: The First 30 Years
• Nonlocal Operators, Parabolic-type Equations, and Ultrametric Random Walks
• On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion
• Harmonic Analysis on the Adèle Ring of $\Q$
• Invariant Parabolic equations and Markov process on Adéles
• A Heat Equation on some Adic Completions of Q and Ultrametric Analysis
• The Non-Archimedean Stochastic Heat Equation driven by Gaussian Noise
• Non-Archimedean Reaction-Ultradiffusion Equations and Complex Hierarchic Systems