Asymptotic behavior of the stochastic heat equation over large intervals
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918134440525824 |
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| author | Candil, David Dalang, Robert C. Solé, Marta Sanz |
| author_facet | Candil, David Dalang, Robert C. Solé, Marta Sanz |
| contents | We consider a nonlinear stochastic heat equation on $[0,T]\times [-L,L]$, driven by a space-time white noise $W$, with a given initial condition $u_0: \mathbb{R} \to \mathbb{R}$ and three different types of (vanishing) boundary conditions: Dirichlet, Mixed and Neumann. We prove that as $L\to\infty$, the random field solution at any space-time position converges in the $L^p(Ω)$-norm ($p\ge 1$) to the solution of the stochastic heat equation on $\mathbb{R}$ (with the same initial condition $u_0$), and we determine the (near optimal) rate of convergence. The proof relies on estimates of differences between the corresponding Green's functions on $[-L, L]$ and the heat kernel on $\mathbb{R}$, and on a space-time version of a Gronwall-type lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of the stochastic heat equation over large intervals Candil, David Dalang, Robert C. Solé, Marta Sanz Probability Primary 60H15, 35R60, Secondary 60H20, 35K05, 35K08, 35A35 We consider a nonlinear stochastic heat equation on $[0,T]\times [-L,L]$, driven by a space-time white noise $W$, with a given initial condition $u_0: \mathbb{R} \to \mathbb{R}$ and three different types of (vanishing) boundary conditions: Dirichlet, Mixed and Neumann. We prove that as $L\to\infty$, the random field solution at any space-time position converges in the $L^p(Ω)$-norm ($p\ge 1$) to the solution of the stochastic heat equation on $\mathbb{R}$ (with the same initial condition $u_0$), and we determine the (near optimal) rate of convergence. The proof relies on estimates of differences between the corresponding Green's functions on $[-L, L]$ and the heat kernel on $\mathbb{R}$, and on a space-time version of a Gronwall-type lemma. |
| title | Asymptotic behavior of the stochastic heat equation over large intervals |
| topic | Probability Primary 60H15, 35R60, Secondary 60H20, 35K05, 35K08, 35A35 |
| url | https://arxiv.org/abs/2509.02504 |