Nonlocal Operators, Parabolic-type Equations, and Ultrametric Random Walks

Abstract

In this article we introduce a new type of nonlocal operators and study the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated to these operators. Some of these equations are the p-adic master equations of certain models of complex systems introduced by Avetisov et al. The fundamental solutions of these parabolic-type equations are transition functions of random walks on the n-dimensional vector space over the field of p-adic numbers. We study some properties of these random walks, including the first passage time.

arXiv commentDefinition 7.1 was corrected. Two additional references were added

References in corpus (8)

• p-Adic Mathematical Physics
• p-Adic Models of Ultrametric Diffusion Constrained by Hierarchical Energy Landscapes
• p-Adic description of characteristic relaxation in complex systems
• p-Adic pseudodifferential operators and p-adic wavelets
• First Passage Time Distribution and Number of Returns for Ultrametric Random Walk
• Parabolic Type Equations and Markov Stochastic Processes on Adeles
• Pseudodifferential operators on ultrametric spaces and ultrametric wavelets
• Taibleson Operators, p-adic Parabolic Equations and Ultrametric Diffusion

Cited by in corpus (13)

• $p$-Adic Mathematical Physics: The First 30 Years
• Brownian Motion and Finite Approximations of Quantum Systems over Local Fields
• Ultrametric Diffusion, Exponential Landscapes, and the First Passage Time Problem
• Non-Archimedean Pseudo-Differential Operators With Bessel Potentials
• Some classes of non-archimedean radial probability density functions associated with energy landscapes
• Symbols of non-archimedean elliptic pseudo-differential operators, Feller semigroups, Markov transition function and negative definite functions
• The Non-Archimedean Stochastic Heat Equation driven by Gaussian Noise
• P-adic Elliptic Quadratic Forms, Parabolic-Type Pseudodifferential Equations With Variable Coefficients, and Markov Processes
• Non-local Operators, Non-Archimedean Parabolic-type Equations with Variable Coefficients and Markov Processes
• Non-Archimedean pseudodifferential operators and Feller Semigroups
• P-adic Markov process and the problem of the first return over balls
• Non-Archimedean Reaction-Ultradiffusion Equations and Complex Hierarchic Systems
• Transience and recurrence of random walks on percolation clusters in an ultrametric space