Geometrical quantity on random checkerboards on the regular torus
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913410648637440 |
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| author | Gohier, Léa |
| author_facet | Gohier, Léa |
| contents | In the study of the observability of the wave equation (here on $(0,T)\times \mathbb{T}^d$, where $\mathbb{T}^d$ is the d-dimensional torus), a condition naturally emerges as a sufficient observability condition. This condition, which writes $\ell^T\left(ω\right) > 0$, signifies that the smallest time spent by a geodesic in the subset $ω\subset \mathbb{T}^d$ during time $T$ is non-zero. In other words, the subset $ω$ detects any geodesic propagating on the d-dimensional torus during time $T$. Here, the subset $ω$ is randomly defined by drawing a grid of $n^d$, $n\in\mathbb{N}$, small cubes of equal size and by adding them to $ω$ with probability $\varepsilon > 0$. In this article, we establish a probabilistic property of the functional $\ell^T$: the random law $\ell^T\left(ω_\varepsilon^n\right)$ converges in probability to $\varepsilon$ as $n \to + \infty$.Considering random subsets $ω_\varepsilon^n$ allows us to construct subsets $ω$ such that $\ell^T\left(ω\right) = |ω|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometrical quantity on random checkerboards on the regular torus Gohier, Léa Probability In the study of the observability of the wave equation (here on $(0,T)\times \mathbb{T}^d$, where $\mathbb{T}^d$ is the d-dimensional torus), a condition naturally emerges as a sufficient observability condition. This condition, which writes $\ell^T\left(ω\right) > 0$, signifies that the smallest time spent by a geodesic in the subset $ω\subset \mathbb{T}^d$ during time $T$ is non-zero. In other words, the subset $ω$ detects any geodesic propagating on the d-dimensional torus during time $T$. Here, the subset $ω$ is randomly defined by drawing a grid of $n^d$, $n\in\mathbb{N}$, small cubes of equal size and by adding them to $ω$ with probability $\varepsilon > 0$. In this article, we establish a probabilistic property of the functional $\ell^T$: the random law $\ell^T\left(ω_\varepsilon^n\right)$ converges in probability to $\varepsilon$ as $n \to + \infty$.Considering random subsets $ω_\varepsilon^n$ allows us to construct subsets $ω$ such that $\ell^T\left(ω\right) = |ω|$. |
| title | Geometrical quantity on random checkerboards on the regular torus |
| topic | Probability |
| url | https://arxiv.org/abs/2407.01022 |