Well-posedness of Fractional Stochastic p-Laplace Equations Driven by Superlinear Transport Noise
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918127371026432 |
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| author | Wang, Bixiang |
| author_facet | Wang, Bixiang |
| contents | In this paper, we prove the existence and uniqueness of solutions of the fractional p-Laplace equation with a polynomial drift of arbitrary order driven by superlinear transport noise. By the monotone argument, we first prove the existence and uniqueness of solutions of an abstract stochastic differential equation satisfying a fully local monotonicity condition. We then apply the abstract result to the fractional stochastic p-Laplace equation defined in a bounded domain. The main difficulty is to establish the tightness as well as the uniform integrability of a sequence of approximate solutions defined by the Galerkin method. To obtain the necessary uniform estimates, we employ the Skorokhod-Jakubowski representation theorem on a topological space instead of a metric space. Since the strong Skorokhod representation theorem is incorrect even in a complete separable metric space, we pass to the limit of stochastic integrals with respect to a sequence of Wiener processes by a weak convergence argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06766 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of Fractional Stochastic p-Laplace Equations Driven by Superlinear Transport Noise Wang, Bixiang Probability Analysis of PDEs 60F10, 60H15, 37L55 In this paper, we prove the existence and uniqueness of solutions of the fractional p-Laplace equation with a polynomial drift of arbitrary order driven by superlinear transport noise. By the monotone argument, we first prove the existence and uniqueness of solutions of an abstract stochastic differential equation satisfying a fully local monotonicity condition. We then apply the abstract result to the fractional stochastic p-Laplace equation defined in a bounded domain. The main difficulty is to establish the tightness as well as the uniform integrability of a sequence of approximate solutions defined by the Galerkin method. To obtain the necessary uniform estimates, we employ the Skorokhod-Jakubowski representation theorem on a topological space instead of a metric space. Since the strong Skorokhod representation theorem is incorrect even in a complete separable metric space, we pass to the limit of stochastic integrals with respect to a sequence of Wiener processes by a weak convergence argument. |
| title | Well-posedness of Fractional Stochastic p-Laplace Equations Driven by Superlinear Transport Noise |
| topic | Probability Analysis of PDEs 60F10, 60H15, 37L55 |
| url | https://arxiv.org/abs/2506.06766 |