On weak convergence of stochastic wave equation with colored noise on $\mathbb{R}$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917297529028608 |
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| author | Tao, Wenxuan |
| author_facet | Tao, Wenxuan |
| contents | In this paper, we study the following stochastic wave equation on the real line $\partial_t^2 u_α=\partial_x^2 u_α+b\left(u_α\right)+σ\left(u_α\right)η_α$. The noise $η_α$ is white in time and colored in space with a covariance structure $\mathbb{E}[η_α(t,x)η_α(s,y)]=δ(t-s)f_α(x-y)$ where $f_α$ is continuous with respect to $α$ in Fourier mode, see Assumption 1.2. We prove the continuity of the probability measure induced by the solution $u_α$, in terms of $α$, with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also give several examples of $f_α$ such that our theorem applies to. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On weak convergence of stochastic wave equation with colored noise on $\mathbb{R}$ Tao, Wenxuan Probability 60B10, 60H15 In this paper, we study the following stochastic wave equation on the real line $\partial_t^2 u_α=\partial_x^2 u_α+b\left(u_α\right)+σ\left(u_α\right)η_α$. The noise $η_α$ is white in time and colored in space with a covariance structure $\mathbb{E}[η_α(t,x)η_α(s,y)]=δ(t-s)f_α(x-y)$ where $f_α$ is continuous with respect to $α$ in Fourier mode, see Assumption 1.2. We prove the continuity of the probability measure induced by the solution $u_α$, in terms of $α$, with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also give several examples of $f_α$ such that our theorem applies to. |
| title | On weak convergence of stochastic wave equation with colored noise on $\mathbb{R}$ |
| topic | Probability 60B10, 60H15 |
| url | https://arxiv.org/abs/2408.10326 |