Starting from physical motivations and leading to practical applications, this book provides an interdisciplinary perspective on the cutting edge of ultrametric pseudodifferential equations. It shows the ways in which these equations link different fields including mathematics, engineering, and geophysics. In particular, the authors provide a detailed explanation of the geophysical applications of p-adic diffusion equations, useful when modeling the flows of liquids through porous rock. p-adic wavelets theory and p-adic pseudodifferential equations are also presented, along with their connections to mathematical physics, representation theory, the physics of disordered systems, probability, number theory, and p-adic dynamical systems. Material that was previously spread across many articles in journals of many different fields is brought together here, including recent work on the van der Put series technique. This book provides an excellent snapshot of the fascinating field of ultrametric pseudodifferential equations, including their emerging applications and currently unsolved problems.

• Covers not only the mathematical underpinnings, but also the practical applications of ultrametric pseudodifferential equations • Discusses many fascinating interdisciplinary connections • Contains a chapter devoted to the application of p-adic diffusion equations to model flows of fluids (e.g. oil and water) in capillary networks in porous disordered media, particularly useful for geophysicists

### Contents

1. p-adic analysis: essential ideas and results; 2. Ultrametric geometry: cluster networks and buildings; 3. p-adic wavelets; 4. Ultrametricity in the theory of complex systems; 5. Some applications of wavelets and integral operators; 6. p-adic and ultrametric models in geophysics; 7. Recent development of the theory of p-adic dynamical systems; 8. Parabolic-type equations, Markov processes, and models of complex hierarchic systems; 9. Stochastic heat equation driven by Gaussian noise; 10. Sobolev-type spaces and pseudodifferential operators; 11. Non-archimedean white noise, pseudodifferential stochastic equations, and massive Euclidean fields; 12. Heat traces and spectral zeta functions for p-adic laplacians; References; Index.