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

TL;DR

This paper introduces new nonlocal operators linked to p-adic parabolic equations, analyzing their fundamental solutions as transition functions of ultrametric random walks with applications to complex systems.

Contribution

It defines novel nonlocal operators and explores their associated parabolic equations, connecting them to p-adic random walks and complex system models.

Findings

Fundamental solutions serve as transition functions for ultrametric random walks.

Analysis of first passage times for these p-adic random walks.

Establishment of properties of solutions to the associated parabolic equations.

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.