On a Fractional Variant of Linear Birth-Death Process
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912228837425152 |
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| author | Dhillon, Manisha Vishwakarma, Pradeep Kataria, Kuldeep Kumar |
| author_facet | Dhillon, Manisha Vishwakarma, Pradeep Kataria, Kuldeep Kumar |
| contents | We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP) which is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07329 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Fractional Variant of Linear Birth-Death Process Dhillon, Manisha Vishwakarma, Pradeep Kataria, Kuldeep Kumar Probability Primary: 60J27, Secondary: 60J20 We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP) which is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed. |
| title | On a Fractional Variant of Linear Birth-Death Process |
| topic | Probability Primary: 60J27, Secondary: 60J20 |
| url | https://arxiv.org/abs/2502.07329 |