Classical first passage problems for $p$-adic stochastic processes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908548044161024 |
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| author | Bikulov, A. Kh. Zubarev, A. P. |
| author_facet | Bikulov, A. Kh. Zubarev, A. P. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16096 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classical first passage problems for $p$-adic stochastic processes Bikulov, A. Kh. Zubarev, A. P. Mathematical Physics Probability 35S05, 60J76, 82C44 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. |
| title | Classical first passage problems for $p$-adic stochastic processes |
| topic | Mathematical Physics Probability 35S05, 60J76, 82C44 |
| url | https://arxiv.org/abs/2509.16096 |