Classical first passage problems for $p$-adic stochastic processes

Abstract

In this paper we present a comprehensive analysis of the solution of the classical problem of finding the distribution density of a random variable - the first passage time to a given domain by the trajectory of a $p$-adic Markov stochastic process with probability density function satisfying the solution of the Cauchy problem for the Vladimirov equation (a $p$-adic analog of the Kolmogorov-Feller equation with the kernel of the Vladimirov operator) with uniform initial distribution in the unit ball. We consider three equivalent approaches to obtain equations for the distribution density of a random variable - the first passage time to a given domain by a stochastic trajectory. We find a solution to these equations for the distribution density, analyze its properties, and compare them with the properties of the distribution density of a random variable - the first return timeof a stochastic trajectory to the support of the initial distribution. We also solve the problem of finding the number of hittings a given domain and analyze the solution obtained. In conclusion, we discuss a class of problems related to the study of the distribution density of the passage time to a given domain and the return time to the initial domain for other types of $p$-adic Markov stochastic processes.