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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2203.16113 |
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| _version_ | 1866916292110319616 |
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| author | Adams, Zachary P. |
| author_facet | Adams, Zachary P. |
| contents | We study transient patterns appearing in a class of SPDE using the framework of quasi-stationary and quasi-ergodic measures. In particular, we prove the existence and uniqueness of quasi-stationary and quasi-ergodic measures for a class of reaction-diffusion systems perturbed by additive cylindrical noise. We obtain convergence results in $L^2$ and almost surely, and demonstrate an exponential rate of convergence to the quasi-stationary measure in an $L^2$ norm. These results allow us to qualitatively characterize the behaviour of these systems in neighbourhoods of an invariant manifold of the corresponding deterministic systems at some large time $t>0$, conditioned on remaining in the neighbourhood up to time $t$. The approach we take here is based on spectral gap conditions, and is not restricted to the small noise regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_16113 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations Adams, Zachary P. Probability We study transient patterns appearing in a class of SPDE using the framework of quasi-stationary and quasi-ergodic measures. In particular, we prove the existence and uniqueness of quasi-stationary and quasi-ergodic measures for a class of reaction-diffusion systems perturbed by additive cylindrical noise. We obtain convergence results in $L^2$ and almost surely, and demonstrate an exponential rate of convergence to the quasi-stationary measure in an $L^2$ norm. These results allow us to qualitatively characterize the behaviour of these systems in neighbourhoods of an invariant manifold of the corresponding deterministic systems at some large time $t>0$, conditioned on remaining in the neighbourhood up to time $t$. The approach we take here is based on spectral gap conditions, and is not restricted to the small noise regime. |
| title | Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations |
| topic | Probability |
| url | https://arxiv.org/abs/2203.16113 |