Stochastic Extinction, An Average Lyapunov Function Approach
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916338670239744 |
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| author | Foldes, Juraj Stacy, Declan |
| author_facet | Foldes, Juraj Stacy, Declan |
| contents | We study the stability of $\mathcal{M}_0$, an invariant subset of a Markov process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the theory of average Lyapunov functions, we formulate general criteria based on the signs of Lyapunov exponents that guarantee extinction ($X_t \to \mathcal{M}_0$ as $t \to \infty$). Additionally, we provide applications to a stochastic SIS epidemic model on a network with regime-switching, a stochastic differential equation version of the Lorenz system, a general class of discrete-time ecological models, and stochastic Kolmogorov systems. In many examples we improve existing results by removing unnecessary assumptions or providing sharper criteria for the extinction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_19606 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic Extinction, An Average Lyapunov Function Approach Foldes, Juraj Stacy, Declan Probability Classical Analysis and ODEs 37H20, 37H30, 60H10, 60J05, 60J25, 60J70 60F15, 92D25 We study the stability of $\mathcal{M}_0$, an invariant subset of a Markov process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the theory of average Lyapunov functions, we formulate general criteria based on the signs of Lyapunov exponents that guarantee extinction ($X_t \to \mathcal{M}_0$ as $t \to \infty$). Additionally, we provide applications to a stochastic SIS epidemic model on a network with regime-switching, a stochastic differential equation version of the Lorenz system, a general class of discrete-time ecological models, and stochastic Kolmogorov systems. In many examples we improve existing results by removing unnecessary assumptions or providing sharper criteria for the extinction. |
| title | Stochastic Extinction, An Average Lyapunov Function Approach |
| topic | Probability Classical Analysis and ODEs 37H20, 37H30, 60H10, 60J05, 60J25, 60J70 60F15, 92D25 |
| url | https://arxiv.org/abs/2407.19606 |