Pseudodifferential Equations Over Non-Archimedean Spaces

Description

Focusing on p-adic and adelic analogues of pseudodifferential equations, this monograph presents a very general theory of parabolic-type equations and their Markov processes motivated by their connection with models of complex hierarchic systems. The Gelfand-Shilov method for constructing fundamental solutions using local zeta functions is developed in a p-adic setting and several particular equations are studied, such as the p-adic analogues of the Klein-Gordon equation. Pseudodifferential equations for complex-valued functions on non-Archimedean local fields are central to contemporary harmonic analysis and mathematical physics and their theory reveals a deep connection with probability and number theory. The results of this book extend and complement the material presented by Vladimirov, Volovich and Zelenov (1994) and Kochubei (2001), which emphasize spectral theory and evolution equations in a single variable, and Albeverio, Khrennikov and Shelkovich (2010), which deals mainly with the theory and applications of p-adic wavelets. Front Matter....Pages i-xvi p-Adic Analysis: Essential Ideas and Results....Pages 1-11 Parabolic-Type Equations and Markov Processes....Pages 13-41 Non-Archimedean Parabolic-Type Equations with Variable Coefficients....Pages 43-77 Parabolic-Type Equations and Markov Processes on Adeles....Pages 79-125 Fundamental Solutions for Pseudodifferential Operators, and Equations of Schrödinger Type....Pages 127-143 Pseudodifferential Equations of Klein-Gordon Type....Pages 145-165 Final Remarks and Some Open Problems....Pages 167-170 Back Matter....Pages 171-177