Spiking and Resetting
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908728165400576 |
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| author | Bernardin, Cédric Tarsamaev, Vsevolod Vladimirovich |
| author_facet | Bernardin, Cédric Tarsamaev, Vsevolod Vladimirovich |
| contents | We consider a one-dimensional piecewise deterministic Markov process (PDMP) on $[0,1]$ with resetting at $0$ and depending on a small parameter $\varepsilon>0$. In the singular vanishing limit $\varepsilon \to 0$ we prove that the `` resetting '' simple point process associated to the PDMP converges to a point process described by a jump Markov process decorated by ``spikes'' distributed as a time-space Poisson point process with intensity proportional to $dt \otimes x^{-2} dx$. This proves rigorously results appeared previously in \cite{SBDKC25} and also justifies partially a conjecture formulated there. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spiking and Resetting Bernardin, Cédric Tarsamaev, Vsevolod Vladimirovich Probability Dynamical Systems We consider a one-dimensional piecewise deterministic Markov process (PDMP) on $[0,1]$ with resetting at $0$ and depending on a small parameter $\varepsilon>0$. In the singular vanishing limit $\varepsilon \to 0$ we prove that the `` resetting '' simple point process associated to the PDMP converges to a point process described by a jump Markov process decorated by ``spikes'' distributed as a time-space Poisson point process with intensity proportional to $dt \otimes x^{-2} dx$. This proves rigorously results appeared previously in \cite{SBDKC25} and also justifies partially a conjecture formulated there. |
| title | Spiking and Resetting |
| topic | Probability Dynamical Systems |
| url | https://arxiv.org/abs/2512.19517 |