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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.15697 |
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| _version_ | 1866909299295387648 |
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| author | Laurence, Lucie Robert, Philippe |
| author_facet | Laurence, Lucie Robert, Philippe |
| contents | We investigate a class of stochastic chemical reaction networks with $n{\ge}1$ chemical species $S_1$, \ldots, $S_n$, and whose complexes are only of the form $k_iS_i$, $i{=}1$,\ldots, $n$, where $(k_i)$ are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter $N$. A natural hierarchy of fast processes, a subset of the coordinates of $(X_i(t))$, is determined by the values of the mapping $i{\mapsto}k_i$. We show that the scaled vector of coordinates $i$ such that $k_i{=}1$ and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as $N$ gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15697 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales Laurence, Lucie Robert, Philippe Probability We investigate a class of stochastic chemical reaction networks with $n{\ge}1$ chemical species $S_1$, \ldots, $S_n$, and whose complexes are only of the form $k_iS_i$, $i{=}1$,\ldots, $n$, where $(k_i)$ are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter $N$. A natural hierarchy of fast processes, a subset of the coordinates of $(X_i(t))$, is determined by the values of the mapping $i{\mapsto}k_i$. We show that the scaled vector of coordinates $i$ such that $k_i{=}1$ and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as $N$ gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions. |
| title | Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales |
| topic | Probability |
| url | https://arxiv.org/abs/2408.15697 |