Comparison principle for stochastic heat equations driven by $α$-stable white noises
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913291088953344 |
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| author | Wang, Yongjin Yan, Chengxin Zhou, Xiaowen |
| author_facet | Wang, Yongjin Yan, Chengxin Zhou, Xiaowen |
| contents | For a class of non-linear stochastic heat equations driven by $α$-stable white noises for $α\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued càdlàg solutions to such a equation for $p\in(α,2]$ by considering a sequence of approximating stochastic heat equations driven by truncated
$α$-stable white noises obtained by removing the big jumps from the original $α$-stable white noises.
If the $α$-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we prove a comparison theorem on the $L^2$-valued càdlàg solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14818 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Comparison principle for stochastic heat equations driven by $α$-stable white noises Wang, Yongjin Yan, Chengxin Zhou, Xiaowen Probability For a class of non-linear stochastic heat equations driven by $α$-stable white noises for $α\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued càdlàg solutions to such a equation for $p\in(α,2]$ by considering a sequence of approximating stochastic heat equations driven by truncated $α$-stable white noises obtained by removing the big jumps from the original $α$-stable white noises. If the $α$-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we prove a comparison theorem on the $L^2$-valued càdlàg solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function. |
| title | Comparison principle for stochastic heat equations driven by $α$-stable white noises |
| topic | Probability |
| url | https://arxiv.org/abs/2209.14818 |