Approximation of the Lévy-driven stochastic heat equation on the sphere
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908437816803328 |
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| author | Lang, Annika Papini, Andrea Schwarz, Verena |
| author_facet | Lang, Annika Papini, Andrea Schwarz, Verena |
| contents | The stochastic heat equation on the sphere driven by additive Lévy random field is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time, in analogy to the Wiener case. New regularity results are proven for the stochastic heat equation. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. To do so, we restrict to square-integrable random field and optimal strong convergence rates for a given regularity of the initial condition and two different settings of regularity for the driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown. Weak rates for the spectral approximation are discussed. Numerical simulations confirm the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximation of the Lévy-driven stochastic heat equation on the sphere Lang, Annika Papini, Andrea Schwarz, Verena Probability Numerical Analysis 60H35, 65C30, 60H15, 35R60, 33C55, 65M70 The stochastic heat equation on the sphere driven by additive Lévy random field is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time, in analogy to the Wiener case. New regularity results are proven for the stochastic heat equation. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. To do so, we restrict to square-integrable random field and optimal strong convergence rates for a given regularity of the initial condition and two different settings of regularity for the driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown. Weak rates for the spectral approximation are discussed. Numerical simulations confirm the theoretical results. |
| title | Approximation of the Lévy-driven stochastic heat equation on the sphere |
| topic | Probability Numerical Analysis 60H35, 65C30, 60H15, 35R60, 33C55, 65M70 |
| url | https://arxiv.org/abs/2507.05005 |