Mild Solution of Semilinear Rough Stochastic Evolution Equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916110284095488 |
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| author | Liang, Jiahao Tang, Shanjian |
| author_facet | Liang, Jiahao Tang, Shanjian |
| contents | In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $du_{t}=\left[L_{t}u_{t}+f\left(t, u_{t}\right)\right]dt+\left(G_{t}u_{t}+g_{t}\right)d\mathbf{X}_{t}+h\left(t, u_{t}\right)dW_{t}$, where $\left(L_{t}\right)_{t \in \left[0, T\right]}$ is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, $\mathbf{X}$ is a two-step $α$-Hölder rough path with $α\in \left(1/3, 1/2\right]$ and $W$ is a Brownian motion. Existence and uniqueness of the mild solution are given through the stochastic controlled rough path approach and fixed-point argument. As a technical tool to define rough stochastic convolutions, we also develop a general mild stochastic sewing lemma, which is applicable for processes according to a monotone family. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mild Solution of Semilinear Rough Stochastic Evolution Equations Liang, Jiahao Tang, Shanjian Probability 60H15, 60L50 In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $du_{t}=\left[L_{t}u_{t}+f\left(t, u_{t}\right)\right]dt+\left(G_{t}u_{t}+g_{t}\right)d\mathbf{X}_{t}+h\left(t, u_{t}\right)dW_{t}$, where $\left(L_{t}\right)_{t \in \left[0, T\right]}$ is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, $\mathbf{X}$ is a two-step $α$-Hölder rough path with $α\in \left(1/3, 1/2\right]$ and $W$ is a Brownian motion. Existence and uniqueness of the mild solution are given through the stochastic controlled rough path approach and fixed-point argument. As a technical tool to define rough stochastic convolutions, we also develop a general mild stochastic sewing lemma, which is applicable for processes according to a monotone family. |
| title | Mild Solution of Semilinear Rough Stochastic Evolution Equations |
| topic | Probability 60H15, 60L50 |
| url | https://arxiv.org/abs/2401.16815 |