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Classical first passage problems for p-adic stochastic processes

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  • Bikulov, A.Kh.
  • Zubarev, A.P.

Abstract

This paper is a continuation of the research started in our previous paper devoted to the study of the distribution density of a random variable — the first return time to the initial domain of the trajectory of a p-adic Markov stochastic process ξt whose probability density satisfies the Cauchy problem for the Vladimirov equation. Here we present a comprehensive study of the problem of finding the distribution density of another random variable — the first passage time of a given domain by the trajectory of the process ξt. We derive an equation for the distribution density of the first passage time of a random trajectory to the domain. Next, we find a solution of this equation, analyze its properties, and compare them with the properties of the distribution density of the first return time of a stochastic trajectory to the support of the initial distribution. We also solve the problem of finding the distribution of 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.

Suggested Citation

  • Bikulov, A.Kh. & Zubarev, A.P., 2026. "Classical first passage problems for p-adic stochastic processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 690(C).
    • Handle: RePEc:eee:phsmap:v:690:y:2026:i:c:s037843712600186x
    DOI: 10.1016/j.physa.2026.131450
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