Pseudodifferential Equations Over Non-Archimedean Spaces, 1st ed. 2016 Lecture Notes in Mathematics Series, Vol. 2174

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.

Offers a fast introduction to the theory of pseudodifferential equations over non-Archimedean fields and their connections with mathematical physics, probability and number theory

Provides a very general theory of parabolic-type equations and their Markov processes motivated by the models of hierarchic complex systems introduced by Avetisov et al. in around 2000

Combines methods of PDEs, probability and number theory

Includes supplementary material: sn.pub/extras