© 2016

Pseudodifferential Equations Over Non-Archimedean Spaces

  • 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


Part of the Lecture Notes in Mathematics book series (LNM, volume 2174)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. W. A. Zúñiga-Galindo
    Pages 1-11
  3. W. A. Zúñiga-Galindo
    Pages 13-41
  4. W. A. Zúñiga-Galindo
    Pages 79-125
  5. W. A. Zúñiga-Galindo
    Pages 145-165
  6. W. A. Zúñiga-Galindo
    Pages 167-170
  7. Back Matter
    Pages 171-177

About this book


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.


11-XX; 43-XX; 46-XX; 60-XX, 70-XX p-adic functional analysis energy landscapes ultrametricity pseudodifferential operators Schrödinger-type operators Klein-Gordon operators fundamental solutions parabolic-type equations Markov processes Igusa local zeta function

Authors and affiliations

  1. 1.Department of MathematicsCenter for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV)Mexico CityMexico

Bibliographic information


“The book is a valuable contribution to the literature on non-Archimedean analysis and mathematical physics. It will be useful for both specialists and students studying this subject.” (Anatoly N. Kochubei, Mathematical Reviews, October, 2017)