First Passage Time Distribution and Number of Returns for Ultrametric Random Walk

TL;DR

This paper analyzes the behavior of a Markov process on ultrametric spaces, focusing on first passage times and return statistics, providing formulas and asymptotic estimates for these quantities.

Contribution

It introduces a detailed analysis of first passage times and return counts for ultrametric diffusion processes, which are less explored in existing literature.

Findings

Derived a formula for the mean number of returns over time.

Provided asymptotic estimates for large time intervals.

Analyzed the distribution of first passage times in ultrametric spaces.

Abstract

In this paper, we consider a homogeneous Markov process \xi(t;\omega) on an ultrametric space Q_p, with distribution density f(x,t), x in Q_p, t in R_+, satisfying the ultrametric diffusion equation df(x,t)/dt =-Df(x,t). We construct and examine a random variable \tau (\omega) that has the meaning the first passage times. Also, we obtain a formula for the mean number of returns on the interval (0,t] and give its asymptotic estimates for large t.