Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918382220083200 |
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| author | Berger, Quentin Bouchot, Nicolas |
| author_facet | Berger, Quentin Bouchot, Nicolas |
| contents | We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $Ω\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $Ω_N = (NΩ)\cap \mathbb{Z}^d$. We let $P_N$ its transition matrix and we study the properties of its ($L^2$-normalized) principal eigenvector $ϕ_N$, also known as ground state. Under mild assumptions on $Ω$, we give regularity estimates on $ϕ_N$, namely on its $k$-th order differences (or \(k\)-th order derivatives), with a uniform control inside $Ω_N$. We provide a completely probabilistic proof of these estimates: our starting point is a Feynman-Kac representation of $ϕ_N$, combined with gambler's ruin estimates and a new ``multi-mirror'' coupling, which may be of independent interest. We also obtain the same type of estimates for the first eigenfunction $φ_1$ of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting $Ω$. Finally, we take the opportunity to review (and slightly extend) some of the literature on the $L^2$ and uniform convergence of $ϕ_N$ to $φ_1$ in Lipschitz bounded domains of $\mathbb{R}^d$, which can be derived thanks to our estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings Berger, Quentin Bouchot, Nicolas Probability Numerical Analysis Analysis of PDEs 60G50, 65L12 We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $Ω\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $Ω_N = (NΩ)\cap \mathbb{Z}^d$. We let $P_N$ its transition matrix and we study the properties of its ($L^2$-normalized) principal eigenvector $ϕ_N$, also known as ground state. Under mild assumptions on $Ω$, we give regularity estimates on $ϕ_N$, namely on its $k$-th order differences (or \(k\)-th order derivatives), with a uniform control inside $Ω_N$. We provide a completely probabilistic proof of these estimates: our starting point is a Feynman-Kac representation of $ϕ_N$, combined with gambler's ruin estimates and a new ``multi-mirror'' coupling, which may be of independent interest. We also obtain the same type of estimates for the first eigenfunction $φ_1$ of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting $Ω$. Finally, we take the opportunity to review (and slightly extend) some of the literature on the $L^2$ and uniform convergence of $ϕ_N$ to $φ_1$ in Lipschitz bounded domains of $\mathbb{R}^d$, which can be derived thanks to our estimates. |
| title | Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings |
| topic | Probability Numerical Analysis Analysis of PDEs 60G50, 65L12 |
| url | https://arxiv.org/abs/2408.15858 |