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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19588 |
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| _version_ | 1866910920595210240 |
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| author | Ji, Un Cig Kim, Jae Hun |
| author_facet | Ji, Un Cig Kim, Jae Hun |
| contents | In this paper, we prove the unique existence and investigate the $L^{p}$-regularity of solutions to stochastic partial differential equations in Hilbert spaces associated with pseudo-differential operators, driven by Hilbert space-valued Gaussian processes that satisfy certain regularity conditions for the covariance kernels of the Gaussian processes. For our purposes, we develop an $L^{p}$-regularity framework for the solutions to the stochastic partial differential equations associated with pseudo-differential operators. As the main tools, we establish the $p$-th moment maximal inequality for stochastic integrals with respect to a Hilbert space-valued Gaussian process and a Littlewood-Paley type inequality for Banach space-valued functions. Additionally, during our study, we improved the sufficient conditions for Fourier multipliers and examined the covariance kernels for Gaussian processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Partial Differential Equations Associated with Pseudo-Differential Operators and Hilbert Space-Valued Gaussian Processes Ji, Un Cig Kim, Jae Hun Analysis of PDEs Probability 60H15, 60G15, 47G30 In this paper, we prove the unique existence and investigate the $L^{p}$-regularity of solutions to stochastic partial differential equations in Hilbert spaces associated with pseudo-differential operators, driven by Hilbert space-valued Gaussian processes that satisfy certain regularity conditions for the covariance kernels of the Gaussian processes. For our purposes, we develop an $L^{p}$-regularity framework for the solutions to the stochastic partial differential equations associated with pseudo-differential operators. As the main tools, we establish the $p$-th moment maximal inequality for stochastic integrals with respect to a Hilbert space-valued Gaussian process and a Littlewood-Paley type inequality for Banach space-valued functions. Additionally, during our study, we improved the sufficient conditions for Fourier multipliers and examined the covariance kernels for Gaussian processes. |
| title | Stochastic Partial Differential Equations Associated with Pseudo-Differential Operators and Hilbert Space-Valued Gaussian Processes |
| topic | Analysis of PDEs Probability 60H15, 60G15, 47G30 |
| url | https://arxiv.org/abs/2504.19588 |