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.
