Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918100911259648 |
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| author | Jiménez, Juan J. |
| author_facet | Jiménez, Juan J. |
| contents | In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{L}u=\dotξ$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric Lévy noise $\dotξ$. Under general sufficient conditions on the coefficients of the second-order operator $\mathcal{L}$, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fractional Laplacian of power $γ\in (0,\infty)$. In particular, whenever $γ>\tfrac{d}{2}$, the solution admits a continuous modification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise Jiménez, Juan J. Probability In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{L}u=\dotξ$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric Lévy noise $\dotξ$. Under general sufficient conditions on the coefficients of the second-order operator $\mathcal{L}$, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fractional Laplacian of power $γ\in (0,\infty)$. In particular, whenever $γ>\tfrac{d}{2}$, the solution admits a continuous modification. |
| title | Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise |
| topic | Probability |
| url | https://arxiv.org/abs/2507.12656 |