2025/09/19 by A. Kh. Bikulov, Bikulov, A. Kh., А. П. Зубарев +1
Computer Science · Mathematics · #35S05 #60J76 #82C44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Topological and Geometric Data Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2509.16096
openalex publication_date 2025/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.